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1
CHAPTER 1
INTRODUCTION
Modern mechanical and aerospace systems are often very complex and con-
sist of many components interconnected by joints and force elements such as
springs, dampers, and actuators. These systems are referred to, in modern lit-
erature, as multibody systems. Examples of multibody systems are machines,
mechanisms, robotics, vehicles, space structures, and biomechanical systems.
The dynamics of such systems are often governed by complex relationships
resulting from the relative motion and joint forces between the components of
the system. Figure 1 shows a hydraulic excavator, which can be considered as
an example of a multibody system that consists of many components. In the
design of such a tracked vehicle, the engineer must deal with many interrelated
questions with regard to the motion and forces of different components of the
vehicle. Examples of these interrelated questions are the following: What is the
relationship between the forward velocity of the vehicle and the motion of the
track chains? What is the effect of the contact forces between the links of the
track chains and the vehicle components on the motion of the system? What
is the effect of the friction forces between the track chains and the ground on
the motion and performance of the vehicle? What is the effect of the soil–track
interaction on the vehicle dynamics, and how can the soil properties be charac-
terized? How does the geometry of the track chains influence the forces and the
maximum vehicle speed? These questions and many other important questions
must be addressed before the design of the vehicle is completed. To provide
a proper answer to many of these interrelated questions, the development of a
detailed dynamic model of such a complex system becomes necessary. In this
book we discuss in detail the development of the dynamic equations of complex
multibody systems such as the tracked hydraulic excavator shown in Fig. 1. The

2 INTRODUCTION
Figure 1.1 Hydraulic excavator
methods presented in the book will allow the reader to construct systematically
the kinematic and dynamic equations of large-scale mechanical and aerospace
systems that consist of interconnected bodies. The procedures for solving the
resulting coupled nonlinear equations are also discussed.
1.1 COMPUTATIONAL DYNAMICS
The analysis of mechanical and aerospace systems has been carried out in the
past mainly using graphical techniques. Little emphasis was given to compu-
tational methods because of the lack of powerful computing machines. The
primary interest was to analyze systems that consist of relatively small num-
bers of bodies such that the desired solution can be obtained using graphical
techniques or hand calculations. The advent of high-speed computers made it
possible to analyze complex systems that consist of large numbers of bodies
and joints. Classical approaches that are based on Newtonian or Lagrangian
mechanics have been rediscovered and put in a form suitable for the use on
high-speed digital computers.
Despite the fact that the basic theories used in developing many of the com-
puter algorithms currently in use in the analysis of mechanical and aerospace
systems are the same as those of the classical approaches, modern engineers
and scientists are forced to know more about matrix and numerical methods in
order to be able to utilize efficiently the computer technology available. In this
book, classical and modern approaches used in the kinematic and dynamic anal-
ysis of mechanical and aerospace systems that consist of interconnected rigid

1.1 COMPUTATIONAL DYNAMICS 3
bodies are introduced. The main focus of the presentation is on the modeling of
general multibody systems and on developing the relationships that govern the
dynamic motion of these systems. The objective is to develop general method-
ologies that can be applied to a large class of multibody applications. Many
fundamental and computational problems are discussed with the objective of
addressing the merits and limitations of various procedures used in formulating
and solving the equations of motion of multibody systems. This is the sub-
ject of the general area of computational dynamics that is concerned with the
computer solution of the equations of motion of large-scale systems.
The role of computational dynamics is merely to provide tools that can be
used in the dynamic simulation of multibody systems. Various tools can be used
for the analysis and computer simulation of a given system. This is due mainly
to the fact that the form of the kinematic and dynamic equations that govern the
dynamics of a multibody system is not unique. As such, it is important that the
analyst chooses the tool and form of the equations of motion that is most suited
for his or her application. This is not always an easy task and requires familiarity
of the analysts with different formulations and procedures used in the general
area of computational dynamics. The forms of the equations of motion depend
on the choice of the coordinates used to define the system configuration. One
may choose a small or a large number of coordinates. From the computational
viewpoint, there are advantages and drawbacks to each choice. The selection of
a small number of coordinates always leads to a complex system of equations.
Such a choice, however, has the advantage of reducing the number of equa-
tions that need to be solved. The selection of a large number of coordinates, on
the other hand, has the advantage of producing simpler and less coupled equa-
tions at the expense of increasing the problem dimensionality. The main focus
of this book is on the derivation and use of different forms of the equations
of motion. Some formulations lead to a large system of equations expressed in
terms of redundant coordinates, while others lead to a small system of equa-
tions expressed in terms of a minimum set of coordinates. The advantages and
drawbacks of each of these formulations when constrained multibody systems
are considered are discussed in detail.
Generally speaking, multibody systems can be classified as rigid multibody
systems or flexible multibody systems. Rigid multibody systems are assumed to
consist only of rigid bodies. These bodies, however, may be connected by mass-
less springs, dampers, and/or actuators. This means that when rigid multibody
systems are considered, the only components that have inertia are assumed to
be rigid bodies. Flexible multibody systems, on the other hand, contain rigid
and deformable bodies. Deformable bodies have distributed inertia and elastic-
ity which depend on the body deformations. As the deformable body moves, its
shape changes and its inertia and elastic properties become functions of time.
For this reason, the analysis of deformable bodies is more difficult than rigid
body analysis. In this book, the branch of computational dynamics that deals
with rigid multibody systems only is considered. The theory of flexible multi-
body systems is covered by the author in a more advanced text (Shabana, 1998).

4 INTRODUCTION
1.2 MOTION AND CONSTRAINTS
Systems such as machines, mechanisms, robotics, vehicles, space structures,
and biomechanical systems consist of many bodies connected by different types
of joints and different types of force elements, such as springs, dampers, and
actuators. The joints are often used to control the system mobility and restrict
the motion of the system components in known specified directions. Using the
joints and force elements, multibody systems are designed to perform certain
tasks; some of these tasks are simple, whereas others can be fairly complex and
may require the use of certain types of mechanical joints as well as sophisti-
cated control algorithms. Therefore, understanding the dynamics of these sys-
tems becomes crucial at the design stage and also for performance evaluation
and design improvements. To understand the dynamics of a multibody system,
it is necessary to study the motion of its components. In this section, some of
the basic concepts and definitions used in the motion description of rigid bodies
are discussed, and examples of joints that are widely used in multibody system
applications are introduced.
Unconstrained Motion A general rigid body displacement is composed of
translations and rotations. The analysis of a pure translational motion is rela-
tively simple and the dynamic relationships that govern this type of motion are
fully understood. The problem of finite rotation, on the other hand, is not a triv-
ial one since large rigid body rotations are sources of geometric nonlinearities.
Figure 2 shows a rigid body, denoted as body i. The general displacement of
Figure 1.2 Rigid body displacement

1.2 MOTION AND CONSTRAINTS 5
this body can be conveniently described in an inertial XYZ coordinate system
by introducing the body X i
Y i
Zi
coordinate system whose origin Oi
is rigidly
attached to a point on the rigid body. The general displacement of the rigid
body can then be described in terms of the translation of the reference point Oi
and also in terms of a set of coordinates that define the orientation of the body
coordinate system with respect to the inertial frame of reference. For instance,
the general planar motion of this body can be described using three independent
coordinates that define the translation of the body along the X and Y axes as
well as its rotation about the Z axis. The two translational components and
the rotation are three independent coordinates since any one of them can be
changed arbitrarily while keeping the other two coordinates fixed. The body
may translate along the X axis while its displacement along the Y axis and its
rotation about the Z axis are kept fixed.
In the spatial analysis, the configuration of an unconstrained rigid body in the
three-dimensional space is identified using six coordinates. Three coordinates
describe the translations of the body along the three perpendicular axes X, Y,
and Z, and three coordinates describe the rotations of the body about these three
axes. These again are six independent coordinates, since they can be varied
arbitrarily.
Mechanical Joints Mechanical systems, in general, are designed for spe-
cific operations. Each of them has a topological structure that serves a certain
purpose. The bodies in a mechanical system are not free to have arbitrary dis-
placements because they are connected by joints or force elements. While a
force element such as springs and dampers may significantly affect the motion
of the bodies in one or more directions, such an element does not completely
prevent motion in these directions. As a consequence, a force element does not
reduce the number of independent coordinates required to describe the config-
uration of the system. On the other hand, mechanical joints as shown in Fig.
3 are used to allow motion only in certain directions. The joints reduce the num-
Figure 1.3 Mechanical joints

6 INTRODUCTION
Figure 1.4 Cam and gear systems
ber of independent coordinates of the system since they prevent motion in some
directions. Figure 3a shows a prismatic (translational) joint that allows only
relative translation between the two bodies i and j along the joint axis. The use
of this joint eliminates the freedom of body i to translate relative to body j in
any other direction except along the joint axis. It also eliminates the freedom of
body i to rotate with respect to body j. Figure 3b shows a revolute (pin) joint
that allows only relative rotation between bodies i and j. This joint eliminates
the freedom of body i to translate with respect to body j. The cylindrical joint
shown in Fig. 3c allows body i to translate and rotate with respect to body j
along and about the joint axis. However, it eliminates the freedom of body i
to translate or rotate with respect to body j along any axis other than the joint
axis. Figure 3d shows the spherical (ball) joint, which eliminates the relative
translations between bodies i and j. This joint provides body i with the freedom
to rotate with respect to body j about three perpendicular axes.
Other types of joints that are often used in mechanical system applications
are cams and gears. Figure 4 shows examples of cam and gear systems. In Fig.
4a, the shape of the cam is designed such that a desired motion is obtained from
the follower when the cam rotates about its axis. Gears, on the other hand, are
used to transmit a certain type of motion (translation or rotary) from one body
to another. The gears shown in Fig. 4b are used to transmit rotary motion from
one shaft to another. The relationship between the rate of rotation of the driven
gear to that of the driver gear depends on the diameters of the base circles of
the two gears.
1.3 DEGREES OF FREEDOM
A mechanical system may consist of several bodies interconnected by differ-
ent numbers and types of joints and force elements. The degrees of freedom

1.3 DEGREES OF FREEDOM 7
Figure 1.5 Slider crank mechanism
of a system are defined to be the independent coordinates that are required to
describe the configuration of the system. The number of degrees of freedom
depends on the number of bodies and the number and types of joints in the
system. The slider crank mechanism shown in Fig. 5 is used in several engi-
neering applications, such as automobile engines and pumps. The mechanism
consists of four bodies: body 1 is the cylinder frame, body 2 is the crankshaft,
body 3 is the connecting rod, and body 4 is the slider block, which represents the
piston. The mechanism has three revolute joints and one prismatic joint. While
this mechanism has several bodies and several joints, it has only one degree
of freedom; that is, the motion of all bodies in this system can be controlled
and described using only one independent variable. In this case, one needs only
one force input (a motor or an actuator) to control the motion of this mecha-
nism. For instance, a specified input rotary motion to the crankshaft produces
a desired rectilinear motion of the slider block. If the rectilinear motion of the
slider block is selected to be the independent variable, the force that acts on
the slider block can be chosen such that a desired output rotary motion of the
crankshaft OA can be achieved. Similarly, two force inputs are required in order
to be able to control the motion of a mechanical system that has two degrees of
freedom, and n force inputs are required to control the motion of an n-degree-
of-freedom mechanical system.
Figure 6a shows another example of a simple planar mechanism called the
four-bar mechanism. This mechanism, which has only one degree of freedom,
is used in many industrial and technological applications. The motion of the
Figure 1.6 Four-bar mechanism

8 INTRODUCTION
links of the four-bar mechanism can be controlled by using one force input,
such as driving the crankshaft OA using a motor located at point O. A desired
motion trajectory on the coupler link AB can be obtained by selecting the proper
dimensions of the links of the four-bar mechanism. Figure 6b shows the motion
of the center of the coupler AB when the crankshaft OA of the mechanism
shown in Fig. 6a rotates one complete cycle. Different motion trajectories can
be obtained by using different dimensions.
Another one-degree-of-freedom mechansm is the Peaucellier mechanism,
shown in Fig. 7. This mechanism is designed to generate a straight-line path.
The geometry of this mechanism is such that BC c BP c EC c EP and AB c
AE. Points A, C, and P should always lie on a straight line passing through
A. The mechanism always satisfies the condition AC × AP c c, where c is a
constant called the inversion constant. In case AD c CD, point P should follow
an exact straight line.
The majority of mechanism systems form single-degree-of-freedom closed
kinematic chains, in which each member is connected to at least two other mem-
bers. Robotic manipulators as shown in Fig. 8 are examples of multidegree-of-
freedom open-chain systems. Robotic manipulators are designed to synthesize
some aspects of human functions and are used in many applications, such as
welding, painting, material transfers, and assembly tasks. Some of these applica-
tions require high precision and consequently, sophisticated sensors and control
systems are used.
While the number of degrees of freedom of a system is unique and depends
on the system topological structure, the set of degrees of freedom is not unique,
as demonstrated previously by the slider crank mechanism. For this simple
mechanism, the rotation of the crankshaft or the translation of the slider block
Figure 1.7 Peaucellier mechanism

1.4 KINEMATIC ANALYSIS 9
Figure 1.8 Robotic manipulators
can be considered as the system degree of freedom. Depending on the choice
of the degree of freedom, a motor or an actuator can be used to drive the mech-
anism. In the design and control of multibody systems, precise knowledge of
the system degrees of freedom is crucial for motion generation and control. The
number and type of degrees of freedom define the numbers and types of motors
and actuators that must be used at the joints to drive and control the motion
of the multibody system. In Chapter 3, simple criteria are provided for deter-
mining the number of degrees of freedom of multibody systems. These criteria
depend on the number of bodies in the system as well as the number and type
of the joints. When the complexity of the system increases, the identification
of the system degrees of freedom using the simple criteria can be misleading.
For this reason, a numerical procedure for identifying the degrees of freedom
of complex multibody systems is presented in Chapter 6.
1.4 KINEMATIC ANALYSIS
In kinematic analysis we are concerned with the geometric aspects of the
motion of the bodies regardless of the forces that produce this motion. In the
classical approaches used in kinematic analysis, the system degrees of freedom
are first identified. Kinematic relationships are then developed and expressed
in terms of the system degrees of freedom and their time derivatives. The step
of determining the locations and orientations of the bodies in the mechanical
system is referred to as position analysis. In this first step, all the required dis-
placement variables are determined. The second step in kinematic analysis is
velocity analysis, which is used to determine the respective velocities of the
bodies in the system as a function of the time rate of the degrees of freedom.
This can be achieved by differentiating the kinematic relationships obtained
from position analysis. Once the displacements and velocities are determined,
one can proceed to the third step in kinematic analysis, which is referred to
as acceleration analysis. In acceleration analysis, the velocity relationships are

10 INTRODUCTION
differentiated with respect to time to obtain the respective accelerations of the
bodies in the system.
To demonstrate the three principal steps of kinematic analysis, we consider
the two-link manipulator shown in Fig. 9. This manipulator system has two
degrees of freedom, which can be chosen as the angles v2
and v3
that define
the orientation of the two links. Let l2
and l3
be the lengths of the two links
of the manipulator. The global position of the end effector of the manipulator
is defined in the coordinate system XY by the two coordinates rx and ry. These
coordinates can be expressed in terms of the two degrees of freedom v2
and
v3
as follows:
rx c l2
cos v2
+ l3
cos v3
ry c l2
sin v2
+ l3
sin v3 } (1.1)
Note that the position of any other point on the links of the manipulator can
be defined in the XY coordinate system in terms of the degrees of freedom v2
and v3
. Equation 1 represents the position analysis step. Given v2
and v3
, the
position of the end effector or any other point on the links of the manipulator
can be determined.
The velocity equations can be obtained by differentiating the position rela-
tionships of Eq. 1 with respect to time. This yields
ṙx c −v̇2
l2
sin v2
− v̇3
l3
sin v3
ṙy c v̇2
l2
cos v2
+ v̇3
l3
cos v3 } (1.2)
Figure 1.9 Two-degree-of-freedom robot manipulator

1.4 KINEMATIC ANALYSIS 11
Given the degrees of freedom v2
and v3
and their time derivatives, the velocity
of the end effector can be determined using the preceding kinematic equations.
It can also be shown that the velocity of any other point on the manipulator
can be determined in a similar manner.
By differentiating the velocity equations (Eq. 2), the equations that define
the acceleration of the end effector can be written as follows:
r̈x c −v̈2
l2
sin v2
− v̈3
l3
sin v3
− (v̇2
)2
l2
cos v2
− (v̇3
)2
l3
cos v3
r̈y c v̈2
l2
cos v2
+ v̈l3
cos v3
− (v̇2
)2
l2
sin v2
− (v̇3
)2
l3
sin v3 } (1.3)
Therefore, given the degrees of freedom and their first and second time deriva-
tives, the absolute acceleration of the end effector or the acceleration of any
other point on the manipulator links can be determined.
Note that when the degrees of freedom and their first and second time deriva-
tives are specified, there is no need to write force equations to determine the
system configuration. The kinematic position, velocity, and acceleration equa-
tions are sufficient to define the coordinates, velocities, and accelerations of
all points on the bodies of the multibody system. A system in which all the
degrees of freedom are specified is called a kinematically driven system. If one
or more of the system degrees of freedom are not known, it is necessary to
develop the force equations using the laws of motion in order to determine
the system configuration. Such a system will be referred to in this book as a
dynamically driven system.
In the classical approaches, one may have to rely on intuition to select the
degrees of freedom of the system. If the system has a complex topological struc-
ture or has a large number of bodies, difficulties may be encountered when clas-
sical techniques are used. While these techniques lead to simple relationships
for simple mechanisms, they are not suited for the analysis of a large class of
mechanical system applications. Many of the basic concepts used in the classical
approaches, however, are the same as those used for modern computer techniques.
In Chapter 3, two approaches are discussed for kinematically driven multi-
body systems: the classical and computational approaches. In the classical
approach, which is suited for the analysis of simple systems, it is assumed
that the system degrees of freedom can easily be identified and all the kine-
matic variables can be expressed, in a straightforward manner, in terms of the
degrees of freedom. When more complex systems are considered, the use of
another computer-based method, such as the computational approach, becomes
necessary. In the computational approach, the kinematic constraint equations
that describe mechanical joints and specified motion trajectories are formulated,
leading to a relatively large system of nonlinear algebraic equations that can be
solved using computer and numerical methods. This computational method can
be used as the basis for developing a general-purpose computer program for the
kinematic analysis of a large class of kinematically driven multibody systems,
as discussed in Chapter 3.

12 INTRODUCTION
1.5 FORCE ANALYSIS
Forces in mechanical systems can be categorized as inertia, external, and joint
forces. Inertia is the property of a body that causes it to resist any effort to
change its motion. Inertia forces, in general, depend on the mass and shape of
the body as well as its velocity and acceleration. If a body is at rest, its inertia
forces are equal to zero. Joint forces are the reaction forces that arise as the
result of the connectivity between different bodies in mechanical systems. These
forces are sometimes referred to as internal forces or constraint forces. Accord-
ing to Newton’s third law, the joint reaction forces acting on two interconnected
bodies are equal in magnitude and opposite in direction. In this book, exter-
nal forces are forces that are not inertia or joint forces. Examples of external
forces are spring and damper forces, motor torques, actuator forces, and gravity
forces.
While in kinematics we are concerned only with motion without regard to
the forces that cause it, in dynamic analysis we are interested in the motion
and the forces that produce it. Unlike the case of static or kinematic analysis,
where only algebraic equations are used, in dynamic analysis, the motion of a
mechanical system is governed by second-order differential equations. Several
techniques are discussed in this book for the dynamic analysis of mechanical
systems that consist of interconnected rigid bodies. Only the reader’s familiarity
with Newton’s second law is assumed for understanding the developments pre-
sented in later chapters. This law states that the force that acts on a particle is
equal to the rate of change of momentum of the particle. Newton’s second law,
with Euler’s equations that govern the rotation of the rigid body, leads to the
dynamic conditions for the rigid bodies. D’Alembert’s principle, which implies
that inertia forces can be treated the same as applied forces, can be used to
obtain the powerful principle of virtual work. Lagrange used this principle as
a starting point to derive his dynamic equation, which is expressed in terms of
scalar energy quantities. D’Alembert’s principle, the principle of virtual work,
and Lagrange’s equation are discussed in detail in Chapters 4 and 5.
1.6 DYNAMIC EQUATIONS AND THEIR DIFFERENT FORMS
Depending on the number of coordinates selected to define the configuration of
a mechanical system, different equation structures can be obtained and different
solution procedures can be adopted. Some of the formulations lead to equations
that are expressed in terms of the constraint forces, while in other formulations,
the constraint forces are eliminated automatically. For instance, the equations
of motion of a simple system such as the block shown in Fig. 10 can be formu-
lated using a minimum set of independent coordinates or using a redundant set
of coordinates that are not totally independent. Since the system has one degree
of freedom representing the motion in the horizontal direction, one equation
suffices to define the configuration of the block. This equation can simply be

1.6 DYNAMIC EQUATIONS AND THEIR DIFFERENT FORMS 13
Figure 1.10 Forms of the equations of motion
written as
mẍ c F (1.4)
where m is the mass of the block, x is the block coordinate, and F is the force act-
ing on the block. Note that when the force is given, the preceding equation can
be solved for the acceleration. We also note that the preceding equation does not
include reaction forces since this equation describes motion in terms of the degree
of freedom. As we will see in subsequent chapters, it is always possible to obtain
a set of dynamic equations which do not include any constraint forces when the
degrees of freedom are used. The principle of virtual work in dynamics represents
a powerful tool that enables us systematically to formulate a set of dynamic equa-
tions of constrained multibody systems such that these equations do not include
constraint forces. This principle is discussed in detail in Chapter 5.
Another approach that can be used to formulate the equations of motion of
the simple system shown in Fig. 10 is to use redundant coordinates. For exam-
ple, we may choose to describe the dynamics of the block using the following
two equations:
mẍ c F
mÿ c N − mg } (1.5)
where y is the coordinate of the block in the vertical direction, N is the reaction
force due to the constraint imposed on the motion of the block, and g is the
gravity constant. If the force F is given, the preceding two equations have three
unknowns: two acceleration components and the reaction force N. For this rea-
son, another equation is needed to be able to solve for the three unknowns. The
third equation is simply the equation of the constraint imposed on the motion
of the block in the vertical direction. This equation can be written as
y c c (1.6)
where c is a constant. This algebraic equation along with the two differential equa-
tions of motion (Eq. 5) form a system of algebraic and differential equations that

14 INTRODUCTION
can be solved for all the coordinates and forces. Here we obtained a larger sys-
tem expressed in terms of a set of redundant coordinates since the y coordinate is
not a degree of freedom. As we will see in this book, use of the redundant system
can have computational advantages and can also increase the generality and flexi-
bility of the formulation used. For this reason, many general-purpose multibody
computer programs use formulations that employ redundant coordinates. There
are, however, several general observations with regard to the use of redundant
coordinates. Using our simple system, we note that the number of independent
constraint (reaction) forces is equal to the number of coordinates used minus the
number of the system degrees of freedom. We also note that the number of inde-
pendent reaction forces is equal to the number of constraint equations. As we will
see in subsequent chapters, this is always the case regardless of the complexity of
the system analyzed, and the elimination of a reaction force can be equivalent to
the elimination of a dependent coordinate or a constraint equation. In our example
we have one reaction force (N) and one constraint equation (y c c).
When the equations of motion are formulated in terms of the system degrees
of freedom only, one obtains differential equations that can be solved using a
simpler numerical strategy. When the equations of motion are formulated in
terms of redundant coordinates, a more elaborate numerical scheme must be
used to solve the resulting system of algebraic and differential equations. These
algebraic and differential equations for most multibody systems are coupled and
highly nonlinear. Direct numerical integration methods are used to solve for the
system coordinates and velocities, and iterative numerical procedures are used
to check on the violation of the constraint equations. This subject is discussed in
more detail in Chapter 6, in which the Lagrangian formulation of the equations
of motion is introduced. In this formulation, a symmetric structure of equations
of motion expressed in terms of redundant coordinates and constraint forces is
presented. To obtain this symmetric structure, the concept of generalized con-
straint forces, which are expressed in terms of multipliers known as Lagrange
multipliers, is introduced.
The simple example of the one-degree-of-freedom block discussed in this sec-
tion alludes to some of the fundamental issues in computational dynamics. How-
ever, the equations of motion of multibody mechanical systems are not likely to
be as simple as the equations of the block due to the geometric nonlinearities and
the kinematic constraints. As the complexity of the system topology increases, the
dimensionality and nonlinearity increase. Computational methods for modeling
complex and nonlinear multibody systems are discussed in Chapter 6.
1.7 FORWARD AND INVERSE DYNAMICS
In studying the dynamics of mechanical systems, there are two different types
of analysis that can be performed. These are inverse and forward dynamics.
In inverse dynamics, the motion trajectories of all the system degrees of free-
dom are specified and the objective is to determine the forces that produce this

1.7 FORWARD AND INVERSE DYNAMICS 15
motion. This type of analysis requires only the solution of systems of algebraic
equations. There is no need in this type of analysis for the use of numerical inte-
gration methods since the position coordinates, velocities, and accelerations of
the system are known. In the case of the forward dynamics, however, the forces
that produce the motion are given and the objective is to determine the position
coordinates, velocities, and accelerations. In this type of analysis, the acceler-
ations are first determined using the laws of motion. These accelerations must
then be integrated to determine the coordinates and velocities. In most appli-
cations, a closed-form solution is difficult to obtain and, therefore, one must
resort to direct numerical integration methods.
The difference between forward and inverse dynamics can be explained
using a simple example. Consider a mass m which moves only in the hori-
zontal direction with displacement x as the result of the application of a force
F. The equation of motion of the mass is
mẍ c F (1.7)
In forward dynamics, the force F is given and the objective is to determine the
motion of the mass as the result of the application of force. In this case, we
first solve for the acceleration as
ẍ c
F
m
Knowing F and m, we integrate the acceleration to determine the velocity. Using
the preceding equation, we have
dẋ
dt
c
F
m
which yields
∫
ẋ
ẋ0
dẋ c
∫
t
0
F
m
dt
where ẋ0 is the initial velocity of the mass. It follows that
ẋ c ẋ0 +
∫
t
0
F
m
dt
If the force F is known as a function of time, the preceding equation can be used
to solve for the velocity of the mass. Having determined the velocity, following
equation can be used to determine the displacement:
dx
dt
c ẋ

16 INTRODUCTION
from which
x c x0 +
∫
t
0
ẋ dt
where x0 is the initial displacement of the mass. It is clear from this simple exam-
ple that one needs two initial conditions: an initial displacement and an initial
velocity, to be able to integrate the acceleration to determine the displacement and
velocity in response to given forces. In the case of simple systems, one may be
able to obtain closed-form solutions for the velocities and displacements. In more
complex systems, integration of the accelerations to determine the velocities and
displacements must be performed numerically as described in Chapter 6.
In inverse dynamics, on the other hand, there is no need for performing inte-
grations: One need only solve a system of algebraic equations. For instance, if
the displacement of the mass is specified as a function of time, one can sim-
ply differentiate the displacement twice to obtain the acceleration and substitute
the result into the equation of motion of the system to determine the force. For
example, if the displacement of the mass is prescribed as
x c A sin qt
where A and q are known constants, the acceleration of the mass can be defined
simply as
ẍ c −q2
A sin qt
Using the equation of motion of the mass (Eq. 7), the force F can be determined
as
F c mẍ c −mq2
A sin qt
This equation determines the force required to produce the prescribed displace-
ment of the mass.
Inverse dynamics is widely used in the design and control of many indus-
trial and technological applications, such as robot manipulators and space struc-
tures. By specifying the task to be performed by the system, the actuator forces
and motor torques required to accomplish this task successfully can be pre-
dicted. Furthermore, different design alternatives and force configurations can
be explored efficiently using the techniques of inverse dynamics.
1.8 PLANAR AND SPATIAL DYNAMICS
The analysis of planar systems can be considered as a special case of spatial
analysis. In spatial analysis, more coordinates are required to describe the con-

1.8 PLANAR AND SPATIAL DYNAMICS 17
figuration of unconstrained body. As mentioned previously, six coordinates
that define the location of a point on the body and the orientation of a coor-
dinate system rigidly attached to the body are required to describe the uncon-
strained motion of a rigid body in space. In planar analysis, only three
coordinates are required, and one of these coordinates suffices to define the
orientation of the body as compared to three orientation coordinates in the
three-dimensional analysis. Furthermore, in the planar analysis, the order of
rotation is commutative since the rotation is performed about the same axis;
that is, two consecutive rotations can be added and the sequence of perform-
ing these rotations is immaterial. This is not the case, however, in three-
dimensional analysis, where three independent rotations can be performed
about three perpendicular axes. In this case, the order of rotation is not in
general commutative, and two consecutive rotations about two different axes
cannot in general be added. This can be demonstrated by using the simple
block example shown in Fig. 11, which illustrates different sequences of rota-
tions for the same block. In Fig. 11a, the block is first rotated 908
about
the Y axis and then 908
about the Z axis. In Fig. 11b, the same rotations in
Figure 1.11 Sequence of rotations

18 INTRODUCTION
reverse order are employed; that is, the block is first rotated 908
about the Z
axis and then 908
about the Y axis. It is clear from the results presented in Figs.
11a and b that a change in the sequence of rotations leads to different final ori-
entations. We then conclude from this simple example that the order of the
finite rotations in the spatial analysis is not commutative, and for this reason,
the finite rotations in the spatial analysis cannot be in general added or treated
as vector quantities. The subject of the three-dimensional rotations is discussed
in more detail in Chapters 7 and 8, where different sets of orientation coordi-
nates are discussed. These sets include Euler angles, Euler parameters, direction
cosines, and Rodriguez parameters. The general dynamic equations that govern
the constrained and unconstrained spatial motion of rigid body systems are also
developed in Chapter 7. This includes the Newton–Euler equations and recur-
sive formulations that are often used in computer-aided analysis of constrained
mechanical systems.
1.9 COMPUTER AND NUMERICAL METHODS
While the analytical techniques of Newton, D’Alembert, and Lagrange were
developed centuries ago, these classical approaches have proven to be suitable
for implementation on high-speed digital computers when used with matrix and
numerical methods. The application of these methods leads to a set of differen-
tial equations that can be expressed in a matrix form and can be solved using
numerical and computer methods. Several numerical algorithms are developed
based on the Newtonian or the Lagrangian approaches. These algorithms, which
utilize matrix and numerical methods, are used to develop general- and special-
purpose computer programs that can be used for the dynamic simulation and
control of multibody systems that consist of interconnected bodies. These pro-
grams allow the user to introduce, in a systematic manner, elastic or damp-
ing elements such as springs and dampers, nonlinear general forcing functions,
and/or nonlinear constraint equations.
The computational efficiency of the computer programs developed for the
dynamic analysis of mechanical systems depends on many factors, such as the
choice of coordinates and the numerical procedure used for solving the dynamic
equations. The choice of the coordinates directly influences the number and the
degree of nonlinearity of the resulting dynamic equations. The use of a rela-
tively small number of coordinates leads to a higher degree of nonlinearity and
more complex dynamic equations. For this reason, in many of the computa-
tional methods developed for the dynamic analysis of mechanical systems, a
larger number of displacement coordinates is used for the sake of generality.
As pointed out previously, the reader will recognize when studying this book
that there are two basic dynamic formulations which are widely used in the
computer simulation of multibody systems. In the first formulation, the con-
straint forces are eliminated from the dynamic equations by expressing these
equations in terms of the system degrees of freedom. Variables that represent

1.9 COMPUTER AND NUMERICAL METHODS 19
joint coordinates are often used as the degrees of freedom in order to be able to
express the system configuration analytically in terms of these degrees of free-
dom. The use of the joint variables has the advantage of reducing the number of
equations and the disadvantage of increasing the nonlinearity and complexity of
the equations. This can be expected since all the information about the system
dynamics must be included in a smaller set of equations. Formulations that use
the joint variables or the degrees of freedom to obtain a minimum set of equa-
tions are referred to in this book as the embedding techniques. The embedding
techniques are also the basis for developing the recursive methods, which are
widely used in the analysis of robot manipulators. The recursive methods are
discussed in Chapter 7.
Another dynamic formulation that is widely used in the computer simula-
tion of multibody systems is the augmented formulation. In this formulation,
the equations of motion are expressed in terms of redundant set of coordinates
that are not totally independent. Because of this redundancy, the kinematic alge-
braic constraint equations that describe the relationship between these coordi-
nates must be formulated. As a result, the constraint forces appear in the final
form of the equations of motion. Clearly, one of the drawbacks of using this
approach is increasing the number of coordinates and equations. Another draw-
back is the complexity of the numerical algorithm that must be used to solve
the resulting system of differential and algebraic equations. Nonetheless, the
augmented formulation has the advantage of producing simple equations that
have a sparse matrix structure; therefore, these equations can be solved effi-
ciently using sparse matrix techniques. Furthermore, the general-purpose multi-
body computer programs based on the augmented formulation tend to be more
user friendly since they allow the user systematic introduction of any nonlin-
ear constraint or force function. In most general-purpose computer programs
based on the augmented formulation, the motion of the bodies in the system is
described using absolute Cartesian and orientation coordinates. In planar analy-
sis, two Cartesian coordinates that define the location of the origin of the body
coordinate system selected, and one orientation coordinate that defines the ori-
entation of this coordinate system in a global inertial frame, are used. In the
spatial analysis, six absolute coordinates are used to define the location and ori-
entation of the body coordinate system. The use of similar sets of coordinates
for all bodies in the system makes it easy for the user to change the model by
adding or deleting bodies and joints and/or introducing nonlinear forcing and
constraint functions.
Both the embedding technique and augmented formulation are discussed in
detail in this book, and several examples will be used to show the structure of
the equations obtained using each formulation. These two methods have been
applied successfully to the analysis, design, and control of many technological
and industrial applications, including vehicles, mechanisms, robot manipula-
tors, machines, space structures, and biomechanical systems. It is hoped that
by studying these two basic formulations carefully, the reader will be able to
make a better choice of the method that is most suited for his or her application.

20 INTRODUCTION
1.10 ORGANIZATION, SCOPE, AND NOTATIONS OF THE BOOK
The purpose of this book is to provide an introduction to the subject of com-
putational dynamics. The goal is to introduce the reader to various dynamic
formulations that can be implemented on the digital computer. The computer
implementation is necessary to be able to study the dynamic motion of large-
scale systems. The general formulations presented in this book can also be used
to develop general-purpose computer codes that can be used in the analysis of
a large class of multibody system applications. The book is organized in eight
chapters, including this introductory chapter.
As the dimensionality and complexity of multibody systems increase, a
knowledge of matrix and numerical methods becomes necessary for under-
standing the theory behind general- and special-purpose multibody computer
programs. For this reason, Chapter 2 is devoted to a brief introduction to the
subject of linear algebra. Matrix and vector operations and identities as well
as methods for the numerical solution of systems of algebraic equations are
discussed. The QR and singular value decompositions, which can be used in
multibody dynamics to determine velocity transformation matrices that relate
the system velocities to the time derivatives of the degrees of freedom, are also
introduced in this chapter.
The kinematics of multibody systems are discussed in Chapter 3. In this
chapter, kinematically driven systems in which all the degrees of freedom are
specified are investigated. For these systems, to define the system configura-
tion, one need only formulate a set of algebraic equations. There is no need to
use the laws of motion since the degrees of freedom and their time derivatives
are known. Two basic approaches are discussed, the classical approach and the
computational approach. The classical approach is suited for the analysis of sys-
tems that consist of small number of bodies and joints, and in which the degrees
of freedom can be identified easily and intuitively. The computational approach,
on the other hand, is suited for the analysis of complex systems and can be used
to develop a general-purpose computer program for the kinematic analysis of
varieties of multibody system applications. Based on a systematic and general
description of the system topology, a general-purpose computer program can be
developed and used to construct nonlinear kinematic relationships between the
variables. This program can also be used to solve these relationships numeri-
cally, in order to determine the system configuration.
Various forms of the dynamic equations are presented in Chapter 4. A simple
Newtonian mechanics approach is used in this chapter to derive these differ-
ent forms and demonstrate the basic differences between them. It is shown in
this chapter that when the equations of motion are derived in terms of a set of
redundant coordinates, the constraint forces appear explicitly in the equations.
This leads to the augmented form of the equations of motion. It is shown in
Chapter 4 that the constraint forces can be eliminated from the system equations
of motion if these equations are expressed in terms of the degrees of freedom.
This procedure is referred to in this book as the embedding technique.

1.10 ORGANIZATION, SCOPE, AND NOTATIONS OF THE BOOK 21
Although as demonstrated in Chapter 4, the embedding technique can be
applied in the framework of Newtonian mechanics, the result of this technique
can be obtained more elegantly by using the principle of virtual work. This
principle can be used to eliminate the constraint forces systematically and obtain
a minimum set of dynamic equations expressed in terms of the system degrees
of freedom. The concepts of the virtual work and generalized forces that are
necessary for the application of the virtual work principle, Lagrange’s equation,
and the Hamiltonian formulation are among the topics discussed in Chapter
5. The chapter concludes by examining the relationship between the principle
of virtual work and the Gaussian elimination used in the solution of algebraic
systems of equations.
The analytical methods presented in Chapter 5 are used as the foundation
for the computational approaches discussed in Chapter 6. A computer-based
embedding technique and an augmented formulation suitable for the analy-
sis of large-scale constrained multibody systems are introduced. The impor-
tant concepts of the generalized constraint forces and Lagrange multipliers are
discussed. Numerical algorithms for solving the differential and algebraic equa-
tions of multibody systems are also presented in Chapter 6. It is important to
point out that the basic methods presented in Chapter 6 are not different from
the methods presented in Chapters 4 and 5 except for using a certain set of
coordinates that serves our computational goals.
In Chapters 3 through 6, planar examples are used to focus on the main
concepts and the development of the basic methods without delving into the
details of the three-dimensional motion. The analysis of the spatial motion
is presented in Chapter 7. In this chapter, methods for describing the three-
dimensional rotations are developed, and the concept of angular velocity in
spatial analysis is introduced. The three-dimensional form of the equations of
motion is presented in terms of the generalized coordinates and used to obtain
the known Newton–Euler equations. Formulations of the algebraic constraint
equations of several spatial joints, such as the revolute, prismatic, cylindri-
cal, and universal joints, are discussed. The use of Newton–Euler equations
to develop a recursive formulation for multibody systems is also demonstrated
in Chapter 7.
In Chapter 8, special topics are discussed. These topics include gyroscopic
motion and various sets of parameters that can be used to define the orien-
tation of the rigid body in space. These parameters include Euler parameters,
Rodriguez parameters, and the quaternions.
It is important that readers become familiar with the multibody notations
used in this book as described in the preface in order to follow the develop-
ments presented in different chapters. Boldface letters are used to indicate vec-
tors or matrices. Superscripts are used to indicate body numbers. To distinguish
between a superscript that indicates the body number and the power, parenthe-
ses are used whenever a quantity is raised to a certain power. For example, (l5
)3
is a scalar l associated with body 5 raised to the power of 3.

22
CHAPTER 2
LINEAR ALGEBRA
Vector and matrix concepts have proved indispensable in the development of
the subject of dynamics. The formulation of the equations of motion using the
Newtonian or Lagrangian approach leads to a set of second-order simultaneous
differential equations. For convenience, these equations are often expressed in
vector and matrix forms. Vector and matrix identities can be utilized to provide
much less cumbersome proofs of many of the kinematic and dynamic relation-
ships. In this chapter, the mathematical tools required to understand the devel-
opment presented in this book are discussed briefly. Matrices and matrix oper-
ations are discussed in the first two sections. Differentiation of vector functions
and the important concept of linear independence are discussed in Section 3. In
Section 4, important topics related to three-dimensional vectors are presented.
These topics include the cross product, skew-symmetric matrix representations,
Cartesian coordinate systems, and conditions of parallelism. The conditions of
parallelism are used in this book to define the kinematic constraint equations
of many joints in the three-dimensional analysis. Computer methods for solv-
ing algebraic systems of equations are presented in Sections 5 and 6. Among
the topics discussed in these two sections are the Gaussian elimination, piv-
oting and scaling, triangular factorization, and Cholesky decomposition. The
last two sections of this chapter deal with the QR decomposition and the sin-
gular value decomposition. These two types of decompositions have been used
in computational dynamics to identify the independent degrees of freedom of
multibody systems. The last two sections, however, can be omitted during a
first reading of the book.

2.1 MATRICES 23
2.1 MATRICES
An m × n matrix A is an ordered rectangular array that has m × n elements.
The matrix A can be written in the form
A c (aij) c






a11 a12 · · · a1n
a21 a22 · · · a2n
.
.
.
.
.
.
...
.
.
.
am1 am2 · · · amn






(2.1)
The matrix A is called an m × n matrix since it has m rows and n columns. The
scalar element aij lies in the ith row and jth column of the matrix A. Therefore,
the index i, which takes the values 1, 2, . . . , m, denotes the row number, while
the index j, which takes the values 1, 2, . . . , n denotes the column number.
A matrix A is said to be square if m c n. An example of a square matrix is
A c



3.0 −2.0 0.95
6.3 0.0 12.0
9.0 3.5 1.25



In this example, m c n c 3, and A is a 3 × 3 matrix.
The transpose of an m × n matrix A is an n × m matrix denoted as AT
and
defined as
AT
c






a11 a21 · · · am1
a12 a22 · · · am2
.
.
.
.
.
.
...
.
.
.
a1n a2n · · · amn






(2.2)
For example, let A be the matrix
A c
[2.0 −4.0 −7.5 23.5
0.0 8.5 10.0 0.0 ]
The transpose of A is
AT
c






2.0 0.0
−4.0 8.5
−7.5 10.0
23.5 0.0







24 LINEAR ALGEBRA
That is, the transpose of the matrix A is obtained by interchanging the rows
and columns.
A square matrix A is said to be symmetric if aij c aji. The elements on the
upper-right half of a symmetric matrix can be obtained by flipping the matrix
about the diagonal. For example,
A c



3.0 −2.0 1.5
−2.0 0.0 2.3
1.5 2.3 1.5



is a symmetric matrix. Note that if A is symmetric, then A is the same as its
transpose; that is, A c AT
.
A square matrix is said to be an upper-triangular matrix if aij c 0 for i >
j. That is, every element below each diagonal element of an upper-triangular
matrix is zero. An example of an upper-triangular matrix is
A c






6.0 2.5 10.2 −11.0
0 8.0 5.5 6.0
0 0 3.2 −4.0
0 0 0 −2.2






A square matrix is said to be a lower-triangular matrix if aij c 0 for j > i.
That is, every element above the diagonal elements of a lower-triangular matrix
is zero. An example of a lower-triangular matrix is
A c






6.0 0 0 0
2.5 8.0 0 0
10.2 5.5 3.2 0
−11.0 6.0 −4.0 −2.2






The diagonal matrix is a square matrix such that aij c 0 if i ⬆ j, which
implies that a diagonal matrix has element aii along the diagonal with all other
elements equal to zero. For example,
A c



5.0 0 0
0 1.0 0
0 0 7.0



is a diagonal matrix.

2.2 MATRIX OPERATIONS 25
The null matrix or zero matrix is defined to be a matrix in which all the
elements are equal to zero. The unit matrix or identity matrix is a diagonal
matrix whose diagonal elements are nonzero and equal to 1.
A skew-symmetric matrix is a matrix such that aij c −aji. Note that since aij
c −aji for all i and j values, the diagonal elements should be equal to zero. An
example of a skew-symmetric matrix Ã is
Ã c



0 −3.0 −5.0
3.0 0 2.5
5.0 −2.5 0



It is clear that for a skew-symmetric matrix, Ã
T
c −Ã.
The trace of a square matrix is the sum of its diagonal elements. The trace
of an n × n identity matrix is n, while the trace of a skew-symmetric matrix is
zero.
2.2 MATRIX OPERATIONS
In this section we discuss some of the basic matrix operations that are used
throughout the book.
Matrix Addition The sum of two matrices A and B, denoted by A + B, is
given by
A + B c (aij + bij) (2.3)
where bij are the elements of B. To add two matrices A and B, it is necessary
that A and B have the same dimension; that is, the same number of rows and
the same number of columns. It is clear from Eq. 3 that matrix addition is
commutative, that is,
A + B c B + A (2.4)
Matrix addition is also associative, because
A + (B + C) c (A + B) + C (2.5)
Example 2.1
The two matrices A and B are defined as
A c
[3.0 1.0 −5.0
2.0 0.0 2.0 ], B c
[ 2.0 3.0 6.0
−3.0 0.0 −5.0 ]

26 LINEAR ALGEBRA
The sum A + B is
A + B c
[3.0 1.0 −5.0
2.0 0.0 2.0 ]+ [ 2.0 3.0 6.0
−3.0 0.0 −5.0 ]
c
[ 5.0 4.0 1.0
−1.0 0.0 −3.0 ]
while A − B is
A − B c
[3.0 1.0 −5.0
2.0 0.0 2.0 ]−
[ 2.0 3.0 6.0
−3.0 0.0 −5.0 ]
c
[1.0 −2.0 −11.0
5.0 0.0 7.0 ]
Matrix Multiplication The product of two matrices A and B is another
matrix C, defined as
C c AB (2.6)
The element cij of the matrix C is defined by multiplying the elements of the
ith row in A by the elements of the jth column in B according to the rule
cij c ai1b1j + ai2b2j + · · · + ainbnj
c 冱
冱
冱
k
aikbkj (2.7)
Therefore, the number of columns in A must be equal to the number of rows in
B. If A is an m × n matrix and B is an n × p matrix, then C is an m × p matrix.
In general, AB ⬆ BA. That is, matrix multiplication is not communative. Matrix
multiplication, however, is distributive; that is, if A and B are m × p matrices
and C is a p × n matrix, then
(A + B)C c AC + BC (2.8)
Example 2.2
Let
A c



0 4 1
2 1 1
3 2 1



, B c



0 1
0 0
5 2




Then
AB c



0 4 1
2 1 1
3 2 1






0 1
0 0
5 2



c



5 2
5 4
5 5



The product BA is not defined in this example since the number of columns in B
is not equal to the number of rows in A.
The associative law is valid for matrix multiplications. If A is an m × p
matrix, B is a p × q matrix, and C is a q × n matrix, then
(AB)C c A(BC) c ABC
Matrix Partitioning Matrix partitioning is a useful technique that is fre-
quently used in manipulations with matrices. In this technique, a matrix is
assumed to consist of submatrices or blocks that have smaller dimensions. A
matrix is divided into blocks or parts by means of horizontal and vertical lines.
For example, let A be a 4 × 4 matrix. The matrix A can be partitioned by using
horizontal and vertical lines as follows:
A c







a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . . . . . . . . . .
a41 a42 a43 a44







In this example, the matrix A has been partitioned into four submatrices; there-
fore, we can write A compactly in terms of these four submatrices as
A c
[A11 A12
A21 A22 ]
where
A11 c



a11 a12 a13
a21 a22 a23
a31 a32 a33



, A12 c



a14
a24
a34



,
A21 c [a41 a42 a43], A22 c a44
Apparently, there are many ways by which the matrix A can be partitioned. As

28 LINEAR ALGEBRA
we will see in this book, the way the matrices are partitioned depends on many
factors, including the applications and the selection of coordinates.
Partitioned matrices can be multiplied by treating the submatrices like the
elements of the matrix. To demonstrate this, we consider another matrix B such
that AB is defined. We also assume that B is partitioned as follows:
B c
[B11 B12 B13 B14
B21 B22 B23 B24 ]
The product AB is then defined as follows:
AB c
[A11 A12
A21 A22
][B11 B12 B13 B14
B21 B22 B23 B24
]
c
[A11B11 + A12B21 A11B12 + A12B22 A11B13 + A12B23 A11B14 + A12B24
A21B11 + A22B21 A21B12 + A22B22 A21B13 + A22B23 A21B14 + A22B24
]
When two partitioned matrices are multiplied we must make sure that additions
and products of the submatrices are defined. For example, A11B12 must have
the same dimension as A12B22. Furthermore, the number of columns of the
submatrix Aij must be equal to the number of rows in the matrix Bjk. It is,
therefore, clear that when multiplying two partitioned matrices A and B, we
must have for each vertical partitioning line in A a similarly placed horizontal
partitioning line in B.
Determinant The determinant of an n × n square matrix A, denoted as |A|,
is a scalar defined as
|A| c
|
|
|
|
|
|
|
|
|
|
a11 a12 · · · a1n
a21 a22 · · · a2n
.
.
.
.
.
.
...
.
.
.
an1 an2 · · · ann
|
|
|
|
|
|
|
|
|
|
(2.9)
To be able to evaluate the unique value of the determinant of A, some basic
definitions have to be introduced. The minor Mij corresponding to the element
aij is the determinant formed by deleting the ith row and jth column from the
original determinant |A|. The cofactor Cij of the element aij is defined as
Cij c (−1)i + j
Mij (2.10)
Using this definition, the value of the determinant in Eq. 9 can be obtained in
terms of the cofactors of the elements of an arbitrary row i as follows:
|A| c
n
冱
冱
冱
j c 1
aijCij (2.11)

Clearly, the cofactors Cij are determinants of order n − 1. If A is a 2 × 2 matrix
defined as
A c
[a11 a12
a21 a22 ]
the cofactors Cij associated with the elements of the first row are
C11 c (−1)2
a22 c a22, C12 c (−1)3
a21 c −a21
According to the definition of Eq. 11, the determinant of the 2 × 2 matrix A
using the cofactors of the elements of the first row is
|A| c a11C11 + a12C12 c a11a22 − a12a21
If A is 3 × 3 matrix defined as
A c



a11 a12 a13
a21 a22 a23
a31 a32 a33



the determinant of A in terms of the cofactors of the first row is given by
|A| c
3
冱
冱
冱
j c 1
a1jC1j c a11C11 + a12C12 + a13C13
where
C11 c
|
|
|
|
|
a22 a23
a32 a33
|
|
|
|
|
, C12 c −
|
|
|
|
|
a21 a23
a31 a33
|
|
|
|
|
, C13 c
|
|
|
|
|
a21 a22
a31 a32
|
|
|
|
|
That is, the determinant of A is
|A| c a11
|
|
|
|
|
a22 a23
a32 a33
|
|
|
|
|
− a12
|
|
|
|
|
a21 a23
a31 a33
|
|
|
|
|
+ a13
|
|
|
|
|
a21 a22
a31 a32
|
|
|
|
|
c a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31)
(2.12)
One can show that the determinant of a matrix is equal to the determinant of
its transpose, that is,
|A| c |AT
| (2.13)
and the determinant of a diagonal matrix is equal to the product of the diago-
nal elements. Furthermore, the interchange of any two columns or rows only

30 LINEAR ALGEBRA
changes the sign of the determinant. If a matrix has two identical rows or two
identical columns, the determinant of this matrix is equal to zero. This can be
demonstrated by the example of Eq. 12. For instance, if the second and third
rows are identical, a21 c a31, a22 c a32, and a23 c a33. Using these equalities
in Eq. 12, one can show that the determinant of the matrix A is equal to zero.
More generally, a square matrix in which one or more rows (columns) are linear
combinations of other rows (columns) has a zero determinant. For example,
A c



1 0 −3
0 2 5
1 2 2



and B c



1 0 1
0 2 2
−3 5 2



have zero determinants since in A the last row is the sum of the first two rows
and in B the last column is the sum of the first two columns.
A matrix whose determinant is equal to zero is said to be a singular matrix.
For an arbitrary square matrix, singular or nonsingular, it can be shown that the
value of the determinant does not change if any row or column is added to or
subtracted from another.
Inverse of a Matrix A square matrix A−1
that satisfies the relationship
A−1
A c AA−1
c I (2.14)
where I is the identity matrix, is called the inverse of the matrix A. The inverse
of the matrix A is defined as
A−1
c
Ct
|A|
where Ct is the adjoint of the matrix A. The adjoint matrix Ct is the transpose
of the matrix of the cofactors Cij of the matrix A.
Example 2.3
Determine the inverse of the matrix
A c



1 1 1
0 1 1
0 0 1



Solution. The determinant of the matrix A is equal to 1, that is,
|A| c 1

The cofactors of the elements of the matrix A are
C11 c 1, C12 c 0, C13 c 0,
C21 c −1, C22 c 1, C23 c 0,
C31 c 0, C32 c −1, C33 c 1
The adjoint matrix, which is the transpose of the matrix of the cofactors, is given
by
Ct c



C11 C21 C31
C12 C22 C32
C13 C23 C33



c



1 −1 0
0 1 −1
0 0 1



Therefore,
A−1
c
Ct
|A|
c



1 −1 0
0 1 −1
0 0 1



Matrix multiplications show that
A−1
A c



1 −1 0
0 1 −1
0 0 1






1 1 1
0 1 1
0 0 1



c



1 0 0
0 1 0
0 0 1



c AA−1
If A is the 2 × 2 matrix
A c
[a11 a12
a21 a22 ]
the inverse of A can be written simply as
A−1
c
1
|A| [ a22 −a12
−a21 a11 ]
where |A| c a11a22 − a12a21.
If the determinant of A is equal to zero, the inverse of A does not exist. This
is the case of a singular matrix. It can be verified that
(A−1
)T
c (AT
)−1

32 LINEAR ALGEBRA
which implies that the transpose of the inverse of a matrix is equal to the inverse
of its transpose.
If A and B are nonsingular square matrices, then
(AB)−1
c B−1
A−1
In general, the inverse of the product of square nonsingular matrices A1, A2,
. . . , An − 1, An is
(A1A2 · · · An − 1An)−1
c A−1
n A−1
n − 1 · · · A−1
2 A−1
1
This equation can be used to define the inverse of matrices that arise naturally
in mechanics. One of these matrices that appears in the formulations of the
recursive equations of mechanical systems is
D c








I 0 0 0 · · · 0 0
−D2 I 0 0 · · · 0 0
0 −D3 I 0 · · · 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
.
.
.
0 0 0 0 · · · −Dn
I








The matrix D can be written as the product of n − 1 matrices as follows:
D c











I
−D2 I
0 I
0
...
...
0 I






















I
0 I
−D3 I
0
...
...
0 I











· · ·











I
0 I
0 I
0
...
...
−Dn I












from which
D−1
c











I
0 I
0 I
0
...
...
Dn I






















I
0 I
0
...
...
Dn − 1 I
0 I











· · ·











I
D2 I
0 I
0
...
...
0 I











Therefore, the inverse of the matrix D can be written as
D−1
c











I 0 0 · · · 0
D21 I 0 · · · 0
D32 D31 I · · · 0
D43 D42 D41 · · · 0
.
.
.
.
.
.
.
.
.
...
.
.
.
Dn(n − 1) Dn(n − 2) Dn(n − 3) · · · I











where
Dkr c DkDk − 1 · · · Dk − r + 1
Orthogonal Matrices A square matrix A is said to be orthogonal if
AT
A c AAT
c I
In this case
AT
c A−1

34 LINEAR ALGEBRA
That is, the inverse of an orthogonal matrix is equal to its transpose. An example
of orthogonal matrices is
A c I + ṽ sin v + (1 − cos v)(ṽ)2
(2.15)
where v is an arbitrary angle, ṽ is the skew-symmetric matrix
ṽ c



0 −v3 v2
v3 0 −v1
−v2 v1 0



and v1, v2, and v3 are the components of an arbitrary unit vector v, that is, v c
[v1 v2 v3]T
. While ṽ is a skew-symmetric matrix, (ṽ)2
is a symmetric matrix.
The transpose of the matrix A of Eq. 15 can then be written as
AT
c I − ṽ sin v + (1 − cos v)(ṽ)2
It can be shown that
(ṽ)3
c −ṽ, (ṽ)4
c −(ṽ)2
Using these identities, one can verify that the matrix A of Eq. 15 is an orthog-
onal matrix. In addition to the orthogonality, it can be shown that the matrix A
and the unit vector v satisfy the following relationships:
Av c AT
v c A−1
v c v
In computational dynamics, the elements of a matrix can be implicit or
explicit functions of time. At a given instant of time, the values of the elements
of such a matrix determine whether or not a matrix is singular. For example,
consider the following two matrices, which depend on the three variables f, v,
w:
G c



0 cos f sin v sin f
0 sin f − sin v cos f
1 0 cos v



and
G c



sin v sin w cos w 0
sin v cos w − sin w 0
cos v 0 1




2.3 VECTORS 35
The inverses of these two matrices are given as
G−1
c
1
sin v



− sin f cos v cos f cos v sin v
sin v cos f sin v sin f 0
sin f − cos f 0



and
G
−1
c
1
sin v



sin w cos w 0
sin v cos w − sin v sin w 0
− cos v sin w − cos v cos w sin v



It is clear that these two inverses do not exist if sin v c 0. The reader, however,
can show that the matrix A, defined as
A c GG
−1
is an orthogonal matrix and its inverse does exist regardless of the value
of v.
2.3 VECTORS
An n-dimensional vector a is an ordered set
a c (a1, a2, . . . , an) (2.16)
of n scalars. The scalar ai, i c 1, 2, . . . , n is called the ith component of a.
An n-dimensional vector can be considered as an n × 1 matrix that consists
of only one column. Therefore, the vector a can be written in the following
column form:
a c






a1
a2
.
.
.
an






(2.17)
The transpose of this column vector defines the n-dimensional row vector
aT
c [a1 a2 · · · an]

36 LINEAR ALGEBRA
The vector a of Eq. 17 can also be written as
a c [a1 a2 · · · an]T
(2.18)
By considering the vector as special case of a matrix with only one column or
one row, the rules of matrix addition and multiplication apply also to vectors.
For example, if a and b ar two n-dimensional vectors defined as
a c [a1 a2 · · · an]T
b c [b1 b2 · · · bn]T
then a + b is defined as
a + b c [a1 + b1 a2 + b2 · · · an + bn]T
Two vectors a and b are equal if and only if ai c bi for i c 1, 2, . . . , n.
The product of a vector a and scalar a is the vector
aa c [aa1 aa2 · · · aan]T
(2.19)
The dot, inner, or scalar product of two vectors a c [a1 a2 · · · an]T
and b c
[b1 b2 · · · bn]T
is defined by the following scalar quantity:
a . b c aT
b c [a1 a2 · · · an]






b1
b2
.
.
.
bn






c a1b1 + a2b2 + · · · + anbn (2.20a)
which can be written as
a . b c aT
b c
n
冱
冱
冱
i c 1
aibi (2.20b)
It follows that a . b c b . a.
Two vectors a and b are said to be orthogonal if their dot product is equal
to zero, that is,
a . b c aT
b c 0
The length of a vector a denoted as |a| is defined as the square root of the

2.3 VECTORS 37
dot product of a with itself, that is,
|a| c
f
aTa c [(a1)2
+ (a2)2
+ · · · + (an)2
]1/2
(2.21)
The terms modulus, magnitude, norm, and absolute value of a vector are also
used to denote the length of a vector. A unit vector is defined to be a vector
that has length equal to 1. If â is a unit vector, one must have
|â| c [(â1)2
+ (â2)2
+ · · · + (ân)2
]1/2
c 1
If a c [a1 a2 · · · an]T
is an arbitrary vector, a unit vector â collinear with the
vector a is defined by
â c
a
|a|
c
1
|a|
[a1 a2 · · · an]T
Example 2.4
Let a and b be the two vectors
a c [0 1 3 2]T
, b c [−1 0 2 3]T
Then
a + b c [0 1 3 2]T
+ [−1 0 2 3]T
c [−1 1 5 5]T
The dot product of a and b is
a . b c aT
b c [0 1 3 2]






−1
0
2
3






c 0 + 0 + 6 + 6 c 12
Unit vectors along a and b are
â c
a
|a|
c
1
f
14
[0 1 3 2]T
b̂ c
b
|b|
c
1
f
14
[−1 0 2 3]T
It can be easily verified that |â| c |b̂| c 1.
Differentiation In many applications in mechanics, scalar and vector func-
tions that depend on one or more variables are encountered. An example of a

38 LINEAR ALGEBRA
scalar function that depends on the system velocities and possibly on the system
coordinates is the kinetic energy. Examples of vector functions are the coordi-
nates, velocities, and accelerations that depend on time. Let us first consider a
scalar function f that depends on several variables q1, q2, . . . , and qn and the
parameter t, such that
f c f (q1, q2, . . . qn, t) (2.22)
where q1, q2, . . . , qn are functions of t, that is, qi c qi(t).
The total derivative of f with respect to the parameter t is
d f
dt
c
∂f
∂q1
dq1
dt
+
∂f
∂q2
dq2
dt
+ · · · +
∂f
∂qn
dqn
dt
+
∂f
∂t
which can be written using vector notation as
d f
dt
c
[ ∂f
∂q1
∂f
∂q2
· · ·
∂f
∂qn ]













dq1
dt
dq2
dt
.
.
.
dqn
dt













+
∂f
∂t
(2.23)
This equation can be written as
d f
dt
c
∂f
∂q
dq
dt
+
∂f
∂t
(2.24)
in which ∂f /∂t is the partial derivative of f with respect to t, and
q c [q1 q2 · · · qn]T
∂f
∂q
c f q c
[ ∂f
∂q1
∂f
∂q2
· · ·
∂f
∂qn ] (2.25)
That is, the partial derivative of a scalar function with respect to a vector is a
row vector. If f is not an explicit function of t, ∂f /∂t c 0.
Example 2.5
Consider the function
f (q1, q2, t) c (q1)2
+ 3(q2)3
− (t)2

2.3 VECTORS 39
where q1 and q2 are functions of the parameter t. The total derivative of f with
respect to the parameter t is
d f
dt
c
∂f
∂q1
dq1
dt
+
∂f
∂q2
dq2
dt
+
∂f
∂t
where
∂f
∂q1
c 2q1,
∂f
∂q2
c 9(q2)2
,
∂f
∂t
c −2t
Hence
d f
dt
c 2q1
dq1
dt
+ 9(q2)2 dq2
dt
− 2t
c [2q1 9(q2)2
]





dq1
dt
dq2
dt





− 2t
where ∂f /∂q can be recognized as the row vector
∂f
∂q
c f q c [2q1 9(q2)2
]
Consider the case of several functions that depend on several variables. These
functions can be written as
f 1 c f 1(q1, q2, . . . , qn, t)
f 2 c f 2(q1, q2, . . . , qn, t)
.
.
.
f m c f m(q1, q2, . . . , qn, t)









(2.26)
where qi c qi(t), i c 1, 2, . . . , n. Using the procedure previously outlined in
this section, the total derivative of an arbitrary function f j can be written as
d fj
dt
c
∂f j
∂q
dq
dt
+
∂f j
∂t
j c 1, 2, . . . , m
in which ∂f j/∂q is the row vector
∂f j
∂q
c
[ ∂f j
∂q1
∂f j
∂q2
· · ·
∂f j
∂qn ]

40 LINEAR ALGEBRA
It follows that
df
dt
c













d f1
dt
d f2
dt
.
.
.
d fm
dt













c













∂f 1
∂q1
∂f 1
∂q2
· · ·
∂f 1
∂qn
∂f 2
∂q1
∂f 2
∂q2
· · ·
∂f 2
∂qn
.
.
.
.
.
.
...
.
.
.
∂f m
∂q1
∂f m
∂q2
· · ·
∂f m
∂qn


























dq1
dt
dq2
dt
.
.
.
dqn
dt













+













∂f 1
∂t
∂f 2
∂t
.
.
.
∂f m
∂t













(2.27)
where
f c [ f 1 f 2 · · · f m]T
(2.28)
Equation 27 can also be written as
df
dt
c
∂f
∂q
dq
dt
+
∂f
∂t
(2.29)
where the m × n matrix ∂f/∂q, the n-dimensional vector dq/dt, and the
m-dimensional vector ∂f/∂t can be recognized as
∂f
∂q
c fq c













∂f 1
∂q1
∂f 1
∂q2
· · ·
∂f 1
∂qn
∂f 2
∂q1
∂f 2
∂q2
· · ·
∂f 2
∂qn
.
.
.
.
.
.
...
.
.
.
∂f m
∂q1
∂f m
∂q2
· · ·
∂f m
∂qn













(2.30)
dq
dt
c
[dq1
dt
dq2
dt
· · ·
dqn
dt ]
T
(2.31)
∂f
∂t
c ft c
[∂f 1
∂t
∂f 2
∂t
· · ·
∂f m
∂t ]
T
(2.32)
If the function f j is not an explicit function of the parameter t, then ∂f j/∂t is
equal to zero. Note also that the partial derivative of an m-dimensional vector
function f with respect to an n-dimensional vector q is the m × n matrix fq
defined by Eq. 30.

2.3 VECTORS 41
Example 2.6
Consider the vector function f defined as
f c



f 1
f 2
f 3



c



(q1)2 + 3(q2)3 − (t)2
8(q1)2 − 3t
2(q1)2 − 6q1q2 + (q2)2



The total derivative of the vector function f is
df
dt
c










d f1
dt
d f2
dt
d f3
dt










c



2q1 9(q2)2
16q1 0
(4q1 − 6q2) (2q2 − 6q1)








dq1
dt
dq2
dt





+



−2t
−3
0



where the matrix fq can be recognized as
fq c



2q1 9(q2)2
16q1 0
(4q1 − 6q2) (2q2 − 6q1)



and the vector ft is
∂f
∂t
c ft c [−2t − 3 0]T
In the analysis of mechanical systems, we may also encounter scalar func-
tions in the form
Q c qT
Aq (2.33)
Following a similar procedure to the one outlined previously in this section,
one can show that
∂Q
∂q
c qT
(A + AT
) (2.34)
If A is a symmetric matrix, that is A c AT
, one has
∂Q
∂q
c 2qT
A (2.35)

42 LINEAR ALGEBRA
Linear Independence The vectors a1, a2, . . . , an are said to be linearly
dependent if there exist scalars e1, e2, . . . , en, which are not all zeros, such that
e1a1 + e2a2 + · · · + enan c 0 (2.36)
Otherwise, the vectors a1, a2, . . . , an are said to be linearly independent.
Observe that in the case of linearly independent vectors, not one of these vectors
can be expressed in terms of the others. On the other hand, if Eq. 36 holds, and
not all the scalars e1, e2, . . . , en are equal to zeros, one or more of the vectors
a1, a2, . . . , an can be expressed in terms of the other vectors.
Equation 36 can be written in a matrix form as
[a1 a2 · · · an]






e1
e2
.
.
.
en






c 0 (2.37)
which can also be written as
Ae c 0 (2.38)
in which
A c [a1 a2 · · · an] (2.39)
If the vectors a1, a2, . . . , an are linearly dependent, the system of homogeneous
algebraic equations defined by Eq. 38 has a nontrivial solution. On the other
hand, if the vectors a1, a2, . . . , an are linearly independent vectors, then A must
be a nonsingular matrix since the system of homogeneous algebraic equations
defined by Eq. 38 has only the trivial solution
e c A−1
0 c 0
In the case where the vectors a1, a2, . . . , an are linearly dependent, the square
matrix A must be singular. The number of linearly independent columns in a
matrix is called the column rank of the matrix. Similarly, the number of inde-
pendent rows is called the row rank of the matrix. It can be shown that for any
matrix, the row rank is equal to the column rank is equal to the rank of the
matrix. Therefore, a square matrix that has a full rank is a matrix that has lin-
early independent rows and linearly independent columns. Thus, we conclude
that a matrix that has a full rank is a nonsingular matrix.
If a1, a2, . . . , an are n-dimensional linearly independent vectors, any other
n-dimensional vector can be expressed as a linear combination of these vectors.
For instance, let b be another n-dimensional vector. We show that this vector

2.3 VECTORS 43
has a unique representation in terms of the linearly independent vectors a1, a2,
. . . , an. To this end, we write b as
b c x1a1 + x2a2 + · · · + xnan (2.40)
where x1, x2, . . . , and xn are scalars. In order to show that x1, x2, . . . , and xn
are unique, Eq. 40 can be written as
b c [a1 a2 · · · an]






x1
x2
.
.
.
xn






b c Ax (2.41)
where A is a square matrix defined by Eq. 39 and x is the vector
x c [x1 x2 · · · xn]T
Since the vectors a1, a2, . . . , an are assumed to be linearly independent, the
coefficient matrix A in Eq. 41 has a full row rank, and thus it is nonsingular.
This system of algebraic equations has a unique solution x, which can be written
as
x c A−1
b
That is, an arbitrary n-dimensional vector b has a unique representation in terms
of the linearly independent vectors a1, a2, . . . , an.
A familiar and important special case is the case of three-dimensional vec-
tors. One can show that the three vectors
a1 c



1
0
0



, a2 c



0
1
0



, a3 c



0
0
1



are linearly independent. Any other three-dimensional vector b c [b1 b2 b3]T
can be written in terms of the linearly independent vectors a1, a2, and a3 as
b c b1a1 + b2a2 + b3a3
where the coefficients x1, x2, and x3 can be recognized in this special case as
x1 c b1, x2 c b2, x3 c b3
The coefficients x1, x2, and x3 are called the coordinates of the vector b in the
basis defined by the vectors a1, a2, and a3.

44 LINEAR ALGEBRA
Example 2.7
Show that the vectors
a1 c



1
0
0



, a2 c



1
1
0



, a3 c



1
1
1



are linearly independent. Find also the representation of the vector b c [−1 3 0]T
in terms of the vectors a1, a2, and a3.
Solution. In order to show that the vectors a1, a2, and a3 are linearly independent,
we must show that the relationship
e1a1 + e2a2 + e3a3 c 0
holds only when e1 c e2 c e3 c 0. To show this, we write
e1



1
0
0



+ e2



1
1
0



+ e3



1
1
1



c 0
which leads to
e1 + e2 + e3 c 0
e2 + e3 c 0
e3 c 0
Back substitution shows that
e3 c e2 c e1 c 0
which implies that the vectors a1, a2, and a3 are linearly independent.
To find the unique representation of the vector b in terms of these linearly inde-
pendent vectors, we write
b c x1a1 + x2a2 + x3a3
which can be written in matrix form as
b c Ax
where
A c



1 1 1
0 1 1
0 0 1



, b c



−1
3
0



Hence, the coordinate vector x can be obtained as
x c



x1
x2
x3



c A−1
b c



1 −1 0
0 1 −1
0 0 1






−1
3
0



c



−4
3
0




2.4 THREE-DIMENSIONAL VECTORS 45
2.4 THREE-DIMENSIONAL VECTORS
A special case of n-dimensional vectors is the three-dimensional vector. A three-
dimensional vector a has three components, and can be written as
a c [a1 a2 a3]T
(2.42)
Three-dimensional vectors are important in mechanics because the position,
velocity, and acceleration of a particle or an arbitrary point on a rigid or
deformable body can be described in space using three-dimensional vectors.
Since these vectors are a special case of the more general n-dimensional vec-
tors, the rules of vector additions, dot products, scalar multiplications, and dif-
ferentiations of these vectors are the same as discussed in the preceding section.
Cross Product Consider the three-dimensional vectors a c [a1 a2 a3]T
, and
b c [b1 b2 b3]T
. These vectors can be defined by their components in the three-
dimensional space XYZ. Therefore, the vectors a and b can be written in terms
of their components along the X, Y, and Z axes as
a c a1i + a2 j + a3 k
b c b1i + b2 j + b3 k
where i, j, and k are unit vectors defined along the X, Y, and Z axes, respectively.
The cross or vector product of the vectors a and b is another vector c orthog-
onal to both a and b and is defined as
c c a × b c
|
|
|
|
|
|
|
i j k
a1 a2 a3
b1 b2 b3
|
|
|
|
|
|
|
c (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k (2.43a)
which can also be written as
c c



c1
c2
c3



c a × b c



a2b3 − a3b2
a3b1 − a1b3
a1b2 − a2b1



(2.43b)
This vector satisfies the following orthogonality relationships:
a . c c aT
c c 0
b . c c bT
c c 0

46 LINEAR ALGEBRA
It can also be shown that
c c a × b c −b × a (2.44)
If a and b are parallel vectors, it can be shown that c c a × b c 0. It follows
that a × a c 0. If a and b are two orthogonal vectors, that is, aT
b c 0, it can
be shown that
|c| c |a||b|
The following useful identities can also be verified:
a . (b × c) c (a × b) . c
a × (b × c) c (aT
c)b − (aT
b)c } (2.45)
Example 2.8
Let a and b be the three-dimensional vectors
a c [0 − 5 1]T
b c [1 − 2 3]T
The cross product of a and b is
c c a × b c
|
|
|
|
|
|
|
i j k
a1 a2 a3
b1 b2 b3
|
|
|
|
|
|
|
c
|
|
|
|
|
|
|
i j k
0 −5 1
1 −2 3
|
|
|
|
|
|
|
c −13i + j + 5k
The vector c can then be defined as
c c [−13 1 5]T
It is clear that
cT
a c cT
b c 0
a × b c −b × a
Skew-Symmetric Matrix Representation The vector cross product as
defined by Eq. 43 can be represented using matrix notation. By using Eq. 43b,
one can write a × b as

a × b c



a2b3 − a3b2
a3b1 − a1b3
a1b2 − a2b1



c



0 −a3 a2
a3 0 −a1
−a2 a1 0






b1
b2
b3



(2.46)
a × b c ãb (2.47)
where ã is the skew-symmetric matrix associated with the vector a and defined
as
ã c



0 −a3 a2
a3 0 −a1
−a2 a1 0



(2.48)
Similarly, the cross product b × a can be written in a matrix form as
b × a c −a × b c b̃a (2.49)
where b̃ is the skew-symmetric matrix associated with the vector b and is
defined as
b̃ c



0 −b3 b2
b3 0 −b1
−b2 b1 0



If â is a unit vector along the vector a, it is clear that
â × a c −a × â c 0
It follows that
−ãâ c ãT
â c 0 (2.50)
In some of the developments presented in this book, the constraints that rep-
resent mechanical joints in the system can be expressed using a set of algebraic
equations. Quite often, one encounters a system of equations that can be written

48 LINEAR ALGEBRA
in the following form:
a × x c 0 (2.51)
where a c [a1 a2 a3]T
and x c [x1 x2 x3]T
. Using the notation of the skew
symmetric matrices, Eq. 51 can be written as
ãx c 0 (2.52)
where ã is defined by Eq. 48. Equation 52 leads to the following three algebraic
equations:
a2x3 − a3x2 c 0
a3x1 − a1x3 c 0
a1x2 − a2x1 c 0



(2.53)
These three equations are not independent because, for instance, adding a1/a3
times the first equation to a2/a3 times the second equation leads to the third
equation. That is, the system of equations given by Eq. 51 or equivalently, Eq.
52, has at most two independent equations. This is due primarily to the fact
that the skew-symmetric matrix ã of Eq. 48 is singular and its rank is at most
two.
Example 2.9
Let a and b be the three-dimensional vectors
a c [−1 7 1]T
b c [0 − 3 8]T
Determine the skew-symmetric matrices ã and b̃ associated, respectively, with the
vectors a and b and evaluate the cross product a × b.
Solution. The skew-symmetric matrices ã and b̃ are
ã c



0 −1 7
1 0 1
−7 −1 0



, b̃ c



0 −8 −3
8 0 0
3 0 0



The cross product a × b can be written as
a × b c ãb c



0 −1 7
1 0 1
−7 −1 0






0
−3
8



c



59
8
3




Example 2.10
Solve the system of equations
ãx c 0
where a is the vector
a c [−1 7 1]T
Solution. As pointed out in this section, the system of equations ãx c 0 has only
two independent equations since the rank of the skew-symmetric matrix ã is at
most two. Consequently, this system of equations has a nontrivial solution that can
be determined to within an arbitrary constant. The equation ãx c 0 can be written
explicitly as
a2x3 − a3x2 c 0
a3x1 − a1x3 c 0
a1x2 − a2x1 c 0
Since this system has only two independent equations, we can determine x2 and x3
in terms of x1. This leads to
x2 c
a2
a1
x1, x3 c
a3
a1
x1
This solution satisfies the three algebraic equations, and for a given value of x1,
the other two variables x2 and x3 can be determined. Using the components of the
vector a, we have
x2 c
a2
a1
x1 c −7x1
x3 c
a3
a1
c −x1
Therefore, the solution vector x is
x c



1
−7
−1



x1
Cartesian Coordinate System In spatial dynamics, several sets of orienta-
tion coordinates can be used to describe the three-dimensional rotations. Some
of these orientation coordinates, as will be demonstrated in Chapter 7, lack any
clear physical meaning, making it difficult in many applications to define the
initial configuration of the bodies using these coordinates. One method which
is used in computational dynamics to define a Cartesian coordinate system is
to introduce three points on the rigid body and use the vector cross product to

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Wednesday, 27th.—A beautiful mild morning; the sun shone; the
lake was still, and all the shores reflected in it. I finished my letter to
Mary. Wm. wrote to Stuart. I copied sonnets for him. Mr. Olliff called
and asked us to tea to-morrow. We stayed in the house till the sun
shone more dimly and we thought the afternoon was closing in, but
though the calmness of the Lake was gone with the bright sunshine,
yet it was delightfully pleasant. We found no letter from Coleridge.
One from Sara which we sate upon the wall to read; a sweet long
letter, with a most interesting account of Mr. Patrick. We cooked no
dinner. Sate a while by the fire, and then drank tea at Frank Raty's.
As we went past the Nab I was surprised to see the youngest child
amongst them running about by itself, with a canny round fat face,
and rosy cheeks. I called in. They gave me some nuts. Everybody
surprised that we should come over Grisdale. Paid £1: 3: 3 for letters
come since December 1st. Paid also about 8 shillings at Penrith. The
bees were humming about the hive. William raked a few stones off
the garden, his first garden labour this year. I cut the shrubs. When
we returned from Frank's, Wm. wasted his mind in the Magazines. I
wrote to Coleridge, and Mrs. C., closed the letters up to Samson.
Then we sate by the fire, and were happy, only our tender thoughts
became painful.47
Went to bed at ½ past 11.
Thursday, 28th.—A downright rain. A wet night. Wm. wrote an
epitaph, and altered one that he wrote when he was a boy. It
cleared up after dinner. We were both in miserable spirits, and very
doubtful about keeping our engagements to the Olliffs. We walked
first within view of Rydale then to Lowthwaite, then we went to Mr.
Olliff. We talked a while. Wm. was tired. We then played at cards.
Came home in the rain. Very dark. Came with a lantern. Wm. out of
spirits and tired. He called at ¼ past 3 to know the hour.
Friday, 29th January.—Wm. was very unwell. Worn out with his
bad night's rest. I read to him, to endeavour to make him sleep.
Then I came into the other room, and I read the first book of
Paradise Lost. After dinner we walked to Ambleside.... A heart-
rending letter from Coleridge. We were sad as we could be. Wm.

wrote to him. We talked about Wm.'s going to London. It was a mild
afternoon. There was an unusual softness in the prospects as we
went, a rich yellow upon the fields, and a soft grave purple on the
waters. When we returned many stars were out, the clouds were
moveless, and the sky soft purple, the lake of Rydale calm, Jupiter
behind. Jupiter at least we call him, but William says we always call
the largest star Jupiter. When we came home we both wrote to C. I
was stupefied.
Saturday, January 30th.—A cold dark morning. William chopped
wood. I brought it in a basket.... He asked me to set down the story
of Barbara Wilkinson's turtle dove. Barbara is an old maid. She had
two turtle doves. One of them died, the first year I think. The other
continued to live alone in its cage for nine years, but for one whole
year it had a companion and daily visitor—a little mouse, that used
to come and feed with it; and the dove would carry it and cover it
over with its wings, and make a loving noise to it. The mouse,
though it did not testify equal delight in the dove's company, was yet
at perfect ease. The poor mouse disappeared, and the dove was left
solitary till its death. It died of a short sickness, and was buried
under a tree, with funeral ceremony by Barbara and her maidens,
and one or two others.
On Saturday, 30th, Wm. worked at The Pedlar all the morning. He
kept the dinner waiting till four o'clock. He was much tired....
Sunday, 31st.—Wm. had slept very ill. He was tired. We walked
round the two lakes. Grasmere was very soft, and Rydale was
extremely beautiful from the western side. Nab Scar was just topped
by a cloud which, cutting it off as high as it could be cut off, made
the mountain look uncommonly lofty.48
We sate down a long time
with different plans. I always love to walk that way, because it is the
way I first came to Rydale and Grasmere, and because our dear
Coleridge did also. When I came with Wm., 6 and ½ years ago, it
was just at sunset. There was a rich yellow light on the waters, and
the islands were reflected there. To-day it was grave and soft, but
not perfectly calm. William says it was much such a day as when

Coleridge came with him. The sun shone out before we reached
Grasmere. We sate by the roadside at the foot of the Lake, close to
Mary's dear name, which she had cut herself upon the stone. Wm.
cut at it with his knife to make it plainer.49
We amused ourselves for
a long time in watching the breezes, some as if they came from the
bottom of the lake, spread in a circle, brushing along the surface of
the water, and growing more delicate as it were thinner, and of a
paler colour till they died away. Others spread out like a peacock's
tail, and some went right forward this way and that in all directions.
The lake was still where these breezes were not, but they made it all
alive. I found a strawberry blossom in a rock. The little slender
flower had more courage than the green leaves, for they were but
half expanded and half grown, but the blossom was spread full out.
I uprooted it rashly, and I felt as if I had been committing an
outrage, so I planted it again. It will have but a stormy life of it, but
let it live if it can. We found Calvert here. I brought a handkerchief
full of mosses, which I placed on the chimneypiece when Calvert
was gone. He dined with us, and carried away the encyclopædias.
After they were gone, I spent some time in trying to reconcile myself
to the change, and in rummaging out and arranging some other
books in their places. One good thing is this—there is a nice elbow
place for Wm., and he may sit for the picture of John Bunyan any
day. Mr. Simpson drank tea with us. We paid our rent to Benson....
Monday, February 1st.—Wm. slept badly. I baked bread. William
worked hard at The Pedlar, and tired himself.... There was a purplish
light upon Mr. Olliff's house, which made me look to the other side
of the vale, when I saw a strange stormy mist coming down the side
of Silver How of a reddish purple colour. It soon came on a heavy
rain.... A box with books came from London. I sate by W.'s bedside,
and read in The Pleasures of Hope to him, which came in the box.
He could not fall asleep.
Tuesday, 2nd February.— ... Wm. went into the orchard after
breakfast, to chop wood. We walked into Easedale.... Walked
backwards and forwards between Goody Bridge and Butterlip How.

William wished to break off composition, but was unable, and so did
himself harm. The sun shone, but it was cold. William worked at The
Pedlar. After tea I read aloud the eleventh book of Paradise Lost. We
were much impressed, and also melted into tears. The papers came
in soon after I had laid aside the book—a good thing for my Wm....
Wednesday, 3rd.—A rainy morning. We walked to Rydale for
letters. Found one from Mrs. Cookson and Mary H. It snowed upon
the hills. We sate down on the wall at the foot of White Moss. Sate
by the fire in the evening. Wm. tired, and did not compose. He went
to bed soon, and could not sleep. I wrote to Mary H. Sent off the
letter by Fletcher. Wrote also to Coleridge. Read Wm. to sleep after
dinner, and read to him in bed till ½ past one.
Thursday, 4th.— ... Wm. thought a little about The Pedlar. Read
Smollet's life.
Friday, 5th.—A cold snowy morning. Snow and hail showers. We
did not walk. Wm. cut wood a little. Sate up late at The Pedlar.
Saturday, 6th February.— ... Two very affecting letters from
Coleridge; resolved to try another climate. I was stopped in my
writing, and made ill by the letters.... Wrote again after tea, and
translated two or three of Lessing's Fables.
Sunday, 7th.—A fine clear frosty morning. The eaves drop with
the heat of the sun all day long. The ground thinly covered with
snow. The road black, rocks black. Before night the island was quite
green. The sun had melted all the snow. Wm. working at his poem.
We sate by the fire, and did not walk, but read The Pedlar, thinking
it done; but W. could find fault with one part of it. It was
uninteresting, and must be altered. Poor Wm.!
Monday Morning, 8th February 1802.—It was very windy and
rained hard all the morning. William worked at his poem and I read
a little in Lessing and the grammar. A chaise came past.

After dinner (i.e. we set off at about ½ past 4) we went towards
Rydale for letters. It was a "cauld clash." The rain had been so cold
that it hardly melted the snow. We stopped at Park's to get some
straw round Wm.'s shoes. The young mother was sitting by a bright
wood fire, with her youngest child upon her lap, and the other two
sate on each side of the chimney. The light of the fire made them a
beautiful sight, with their innocent countenances, their rosy cheeks,
and glossy curling hair. We sate and talked about poor Ellis, and our
journey over the Hawes. Before we had come to the shore of the
Lake, we met our patient bow-bent friend, with his little wooden box
at his back. "Where are you going?" said he. "To Rydale for letters."
"I have two for you in my box." We lifted up the lid, and there they
lay. Poor fellow, he straddled and pushed on with all his might; but
we outstripped him far away when we had turned back with our
letters.... I could not help comparing lots with him. He goes at that
slow pace every morning, and after having wrought a hard day's
work returns at night, however weary he may be, takes it all quietly,
and, though perhaps he neither feels thankfulness nor pleasure,
when he eats his supper, and has nothing to look forward to but
falling asleep in bed, yet I daresay he neither murmurs nor thinks it
hard. He seems mechanised to labour. We broke the seal of
Coleridge's letters, and I had light enough just to see that he was
not ill. I put it in my pocket. At the top of the White Moss I took it to
my bosom,—a safer place for it. The sight was wild. There was a
strange mountain lightness, when we were at the top of the White
Moss. I have often observed it there in the evenings, being between
the two valleys. There is more of the sky there than any other place.
It has a strange effect. Sometimes, along with the obscurity of
evening, or night, it seems almost like a peculiar sort of light. There
was not much wind till we came to John's Grove, then it roared right
out of the grove, all the trees were tossing about. Coleridge's letter
somewhat damped us. It spoke with less confidence about France.
Wm. wrote to him. The other letter was from Montagu, with £8.
Wm. was very unwell, tired when he had written. He went to bed
and left me to write to M. H., Montagu, and Calvert, and Mrs.
Coleridge. I had written in his letter to Coleridge. We wrote to

Calvert to beg him not to fetch us on Sunday. Wm. left me with a
little peat fire. It grew less. I wrote on, and was starved. At 2 o'clock
I went to put my letters under Fletcher's door. I never felt such a
cold night. There was a strong wind and it froze very hard. I
gathered together all the clothes I could find (for I durst not go into
the pantry for fear of waking Wm.). At first when I went to bed I
seemed to be warm. I suppose because the cold air, which I had just
left, no longer touched my body; but I soon found that I was
mistaken. I could not sleep from sheer cold. I had baked pies and
bread in the morning. Coleridge's letter contained prescriptions.
N.B.—The moon came out suddenly when we were at John's
Grove, and a star or two besides.
Tuesday.—Wm. had slept better. He fell to work, and made
himself unwell. We did not walk. A funeral came by of a poor woman
who had drowned herself, some say because she was hardly treated
by her husband; others that he was a very decent respectable man,
and she but an indifferent wife. However this was, she had only
been married to him last Whitsuntide and had had very indifferent
health ever since. She had got up in the night, and drowned herself
in the pond. She had requested to be buried beside her mother, and
so she was brought in a hearse. She was followed by some very
decent-looking men on horseback, her sister—Thomas Fleming's wife
—in a chaise, and some others with her, and a cart full of women.
Molly says folks thinks o' their mothers. Poor body, she has been
little thought of by any body else. We did a little of Lessing. I
attempted a fable, but my head ached; my bones were sore with the
cold of the day before, and I was downright stupid. We went to bed,
but not till Wm. had tired himself.
Wednesday, 10th.—A very snowy morning.... I was writing out
the poem, as we hoped for a final writing.... We read the first part
and were delighted with it, but Wm. afterwards got to some ugly
place, and went to bed tired out. A wild, moonlight night.

Thursday, 11th.— ... Wm. sadly tired and working at The
Pedlar.... We made up a good fire after dinner, and Wm. brought his
mattress out, and lay down on the floor. I read to him the life of Ben
Jonson, and some short poems of his, which were too interesting for
him, and would not let him go to sleep. I had begun with Fletcher,
but he was too dull for me. Fuller says, in his Life of Jonson
(speaking of his plays), "If his latter be not so spriteful and vigorous
as his first pieces, all that are old, and all who desire to be old,
should excuse him therein." He says he "beheld" wit-combats
between Shakespeare and Jonson, and compares Shakespeare to an
English man-of-war, Jonson to a great Spanish galleon. There is one
affecting line in Jonson's epitaph on his first daughter—
Here lies to each her parents ruth,
Mary the daughter of their youth.
At six months' end she parted hence,
In safety of her innocence.
Two beggars to-day. I continued to read to Wm. We were much
delighted with the poem of Penshurst.50
Wm. rose better. I was
cheerful and happy. He got to work again.
Friday, 12th.—A very fine, bright, clear, hard frost. Wm. working
again. I recopied The Pedlar, but poor Wm. all the time at work.... In
the afternoon a poor woman came, she said, to beg, ... but she has
been used to go a-begging, for she has often come here. Her father
lived to the age of 105. She is a woman of strong bones, with a
complexion that has been beautiful, and remained very fresh last
year, but now she looks broken, and her little boy—a pretty little
fellow, and whom I have loved for the sake of Basil—looks thin and
pale. I observed this to her. "Aye," says she, "we have all been ill.
Our house was nearly unroofed in the storm, and we lived in it so for
more than a week." The child wears a ragged drab coat and a fur
cap. Poor little fellow, I think he seems scarcely at all grown since
the first time I saw him. William was with me when we met him in a
lane going to Skelwith Bridge. He looked very pretty. He was walking

lazily, in the deep narrow lane, overshadowed with the hedgerows,
his meal poke hung over his shoulder. He said he "was going a
laiting." Poor creature! He now wears the same coat he had on at
that time. When the woman was gone, I could not help thinking that
we are not half thankful enough that we are placed in that condition
of life in which we are. We do not so often bless God for this, as we
wish for this £50, that £100, etc. etc. We have not, however, to
reproach ourselves with ever breathing a murmur. This woman's was
but a common case. The snow still lies upon the ground. Just at the
closing in of the day, I heard a cart pass the door, and at the same
time the dismal sound of a crying infant. I went to the window, and
had light enough to see that a man was driving a cart, which
seemed not to be very full, and that a woman with an infant in her
arms was following close behind and a dog close to her. It was a
wild and melancholy sight. Wm. rubbed his tables after candles were
lighted, and we sate a long time with the windows unclosed, and
almost finished writing The Pedlar; but poor Wm. wore himself out,
and me out, with labour. We had an affecting conversation. Went to
bed at 12 o'clock.
Saturday, 13th.—It snowed a little this morning. Still at work at
The Pedlar, altering and refitting. We did not walk, though it was a
very fine day. We received a present of eggs and milk from Janet
Dockeray, and just before she went, the little boy from the Hill
brought us a letter from Sara H., and one from the Frenchman in
London. I wrote to Sara after tea, and Wm. took out his old
newspapers, and the new ones came in soon after. We sate, after I
had finished the letter, talking; and Wm. read parts of his Recluse
aloud to me....
Sunday, 14th February.—A fine morning. The sun shines out, but
it has been a hard frost in the night. There are some little snowdrops
that are afraid to put their white heads quite out, and a few
blossoms of hepatica that are half-starved. Wm. left me at work
altering some passages of The Pedlar, and went into the orchard.
The fine day pushed him on to resolve, and as soon as I had read a

letter to him, which I had just received from Mrs. Clarkson, he said
he would go to Penrith, so Molly was despatched for the horse. I
worked hard, got the writing finished, and all quite trim. I wrote to
Mrs. Clarkson, and put up some letters for Mary H., and off he went
in his blue spencer, and a pair of new pantaloons fresh from
London.... I then sate over the fire, reading Ben Jonson's Penshurst,
and other things. Before sunset, I put on my shawl and walked out.
The snow-covered mountains were spotted with rich sunlight, a
palish buffish colour.... I stood at the wishing-gate, and when I came
in view of Rydale, I cast a long look upon the mountains beyond.
They were very white, but I concluded that Wm. would have a very
safe passage over Kirkstone, and I was quite easy about him. After
dinner, a little before sunset, I walked out about 20 yards above
Glow-worm Rock. I met a carman, a Highlander I suppose, with four
carts, the first three belonging to himself, the last evidently to a man
and his family who had joined company with him, and who I
guessed to be potters. The carman was cheering his horses, and
talking to a little lass about ten years of age who seemed to make
him her companion. She ran to the wall, and took up a large stone
to support the wheel of one of his carts, and ran on before with it in
her arms to be ready for him. She was a beautiful creature, and
there was something uncommonly impressive in the lightness and
joyousness of her manner. Her business seemed to be all pleasure—
pleasure in her own motions, and the man looked at her as if he too
was pleased, and spoke to her in the same tone in which he spoke
to his horses. There was a wildness in her whole figure, not the
wildness of a Mountain lass, but of the Road lass, a traveller from
her birth, who had wanted neither food nor clothes. Her mother
followed the last cart with a lovely child, perhaps about a year old,
at her back, and a good-looking girl, about fifteen years old, walked
beside her. All the children were like the mother. She had a very
fresh complexion, but she was blown with fagging up the steep hill,
and with what she carried. Her husband was helping the horse to
drag the cart up by pushing it with his shoulder. I reached home,
and read German till about 9 o'clock. I wrote to Coleridge. Went to

bed at about 12 o'clock.... I slept badly, for my thoughts were full of
Wm.
Monday, 15th February.—It snowed a good deal, and was terribly
cold. After dinner it was fair, but I was obliged to run all the way to
the foot of the White Moss, to get the least bit of warmth into me. I
found a letter from C. He was much better, this was very
satisfactory, but his letter was not an answer to Wm.'s which I
expected. A letter from Annette. I got tea when I reached home,
and then set on reading German. I wrote part of a letter to
Coleridge, went to bed and slept badly.
Tuesday, 16th.—A fine morning, but I had persuaded myself not
to expect Wm., I believe because I was afraid of being disappointed.
I ironed all day. He came just at tea time, had only seen Mary H. for
a couple of hours between Eamont Bridge and Hartshorn Tree. Mrs.
C. better. He had had a difficult journey over Kirkstone, and came
home by Threlkeld. We spent a sweet evening. He was better, had
altered The Pedlar. We went to bed pretty soon. Mr. Graham said he
wished Wm. had been with him the other day—he was riding in a
post-chaise and he heard a strange cry that he could not
understand, the sound continued, and he called to the chaise driver
to stop. It was a little girl that was crying as if her heart would burst.
She had got up behind the chaise, and her cloak had been caught by
the wheel, and was jammed in, and it hung there. She was crying
after it, poor thing. Mr. Graham took her into the chaise, and her
cloak was released from the wheel, but the child's misery did not
cease, for her cloak was torn to rags; it had been a miserable cloak
before, but she had no other, and it was the greatest sorrow that
could befall her. Her name was Alice Fell.51
She had no parents, and
belonged to the next town. At the next town, Mr. G. left money with
some respectable people in the town, to buy her a new cloak.
Wednesday, 17th.—A miserable nasty snowy morning. We did not
walk, but the old man from the hill brought us a short letter from
Mary H. I copied the second part of Peter Bell....

Thursday, 18th.—A foggy morning. I copied new part of Peter Bell
in W.'s absence, and began a letter to Coleridge. Wm. came in with
a letter from Coleridge.... We talked together till 11 o'clock, when
Wm. got to work, and was no worse for it. Hard frost.
* * * * * *
Saturday, 20th.— ... I wrote the first part of Peter Bell....
Sunday, 21st.—A very wet morning. I wrote the 2nd prologue to
Peter Bell.... After dinner I wrote the 1st prologue.... Snowdrops
quite out, but cold and winterly; yet, for all this, a thrush that lives in
our orchard has shouted and sung its merriest all day long ...
Monday, 22nd.—Wm. brought me 4 letters to read—from Annette
and Caroline,52
Mary and Sara, and Coleridge.... In the evening we
walked to the top of the hill, then to the bridge. We hung over the
wall, and looked at the deep stream below. It came with a full,
steady, yet a very rapid flow down to the lake. The sykes made a
sweet sound everywhere, and looked very interesting in the twilight,
and that little one above Mr. Olliff's house was very impressive. A
ghostly white serpent line, it made a sound most distinctly heard of
itself. The mountains were black and steep, the tops of some of
them having snow yet visible.
Tuesday, 23rd.— ... When we came out of our own doors, that
dear thrush was singing upon the topmost of the smooth branches
of the ash tree at the top of the orchard. How long it had been
perched on that same tree I cannot tell, but we had heard its dear
voice in the orchard the day through, along with a cheerful
undersong made by our winter friends, the robins. As we came
home, I picked up a few mosses by the roadside, which I left at
home. We then went to John's Grove. There we sate a little while
looking at the fading landscape. The lake, though the objects on the
shore were fading, seemed brighter than when it is perfect day, and
the island pushed itself upwards, distinct and large. All the shores

marked. There was a sweet, sea-like sound in the trees above our
heads. We walked backwards and forwards some time for dear
John's sake, then walked to look at Rydale. Wm. now reading in
Bishop Hall, I going to read German. We have a nice singing fire,
with one piece of wood....
Wednesday, 24th.—A rainy morning. William returned from Rydale
very wet, with letters. He brought a short one from C., a very long
one from Mary. Wm. wrote to Annette, to Coleridge.... I wrote a little
bit to Coleridge. We sent off these letters by Fletcher. It was a
tremendous night of wind and rain. Poor Coleridge! a sad night for a
traveller such as he. God be praised he was in safe quarters. Wm.
went out. He never felt a colder night.
Thursday, 25th.—A fine, mild, gay, beautiful morning. Wm. wrote
to Montagu in the morning.... I reached home just before dark,
brought some mosses and ivy, and then got tea, and fell to work at
German. I read a good deal of Lessing's Essay. Wm. came home
between 9 and 10 o'clock. We sat together by the fire till bedtime.
Wm. not very much tired.
Friday, 26th.—A grey morning till 10 o'clock, then the sun shone
beautifully. Mrs. Lloyd's children and Mrs. Luff came in a chaise,
were here at 11 o'clock, then went to Mrs. Olliff. Wm. and I
accompanied them to the gate. I prepared dinner, sought out Peter
Bell, gave Wm. some cold meat, and then we went to walk. We
walked first to Butterlip How, where we sate and overlooked the
dale, no sign of spring but the red tints of the woods and trees. Sate
in the sun. Met Charles Lloyd near the Bridge.... Mr. and Mrs. Luff
walked home, the Lloyds stayed till 8 o'clock. Wm. always gets on
better with conversation at home than elsewhere. The chaise-driver
brought us a letter from Mrs. H., a short one from C. We were
perplexed about Sara's coming. I wrote to Mary. Wm. closed his
letter to Montagu, and wrote to Calvert and Mrs. Coleridge. Birds
sang divinely to-day. Wm. better.

Sunday, 28th February.—Wm. employed himself with The Pedlar.
We got papers in the morning.
Monday.—A fine pleasant day, we walked to Rydale. I went on
before for the letters, brought two from M. and S. H. We climbed
over the wall and read them under the shelter of a mossy rock. We
met Mrs. Lloyd in going. Mrs. Olliff's child ill. The catkins are
beautiful in the hedges, the ivy is very green. Robert Newton's
paddock is greenish—that is all we see of Spring; finished and sent
off the letter to Sara, and wrote to Mary. Wrote again to Sara, and
Wm. wrote to Coleridge. Mrs. Lloyd called when I was in bed.
Tuesday.53
—A fine grey morning.... I read German, and a little
before dinner Wm. also read. We walked on Butterlip How under the
wind. It rained all the while, but we had a pleasant walk. The
mountains of Easedale, black or covered with snow at the tops, gave
a peculiar softness to the valley. The clouds hid the tops of some of
them. The valley was populous and enlivened with streams....
Wednesday.—I was so unlucky as to propose to rewrite The
Pedlar. Wm. got to work, and was worn to death. We did not walk. I
wrote in the afternoon.
Thursday.—Before we had quite finished breakfast Calvert's man
brought the horses for Wm. We had a deal to do, pens to make,
poems to put in order for writing, to settle for the press, pack up;
and the man came before the pens were made, and he was obliged
to leave me with only two. Since he left me at half-past 11 (it is now
2) I have been putting the drawers into order, laid by his clothes
which he had thrown here and there and everywhere, filed two
months' newspapers and got my dinner, 2 boiled eggs and 2 apple
tarts. I have set Molly on to clean the garden a little, and I myself
have walked. I transplanted some snowdrops—the Bees are busy.
Wm. has a nice bright day. It was hard frost in the night. The Robins
are singing sweetly. Now for my walk. I will be busy. I will look well,
and be well when he comes back to me. O the Darling! Here is one
of his bitter apples. I can hardly find it in my heart to throw it into

the fire.... I walked round the two Lakes, crossed the stepping-
stones at Rydale foot. Sate down where we always sit. I was full of
thought about my darling. Blessings on him. I came home at the
foot of our own hill under Loughrigg. They are making sad ravages
in the woods. Benson's wood is going, and the woods above the
River. The wind has blown down a small fir tree on the Rock, that
terminates John's path. I suppose the wind of Wednesday night. I
read German after tea. I worked and read the L. B., enchanted with
the Idiot Boy. Wrote to Wm. and then went to bed. It snowed when
I went to bed.
Friday.—First walked in the garden and orchard, a frosty sunny
morning. After dinner I gathered mosses in Easedale. I saw before
me sitting in the open field, upon his pack of rags, the old Ragman
that I know. His coat is of scarlet in a thousand patches. When I
came home Molly had shook the carpet and cleaned everything
upstairs. When I see her so happy in her work, and exulting in her
own importance, I often think of that affecting expression which she
made use of to me one evening lately. Talking of her good luck in
being in this house, "Aye, Mistress, them 'at's low laid would have
been proud creatures could they but have seen where I is now, fra
what they thought wud be my doom." I was tired when I reached
home. I sent Molly Ashburner to Rydale. No letters. I was sadly
mortified. I expected one fully from Coleridge. Wrote to William,
read the L. B., got into sad thoughts, tried at German, but could not
go on. Read L. B. Blessings on that brother of mine! Beautiful new
moon over Silver How.
Friday Morning.—A very cold sunshiny frost. I wrote The Pedlar,
and finished it before I went to Mrs. Simpson's to drink tea. Miss S.
at Keswick, but she came home. Mrs. Jameson came in and stayed
supper. Fletcher's carts went past and I let them go with William's
letter. Mr. B. S. came nearly home with me. I found letters from
Wm., Mary, and Coleridge. I wrote to C. Sat up late, and could not
fall asleep when I went to bed.

* * * * * *
Sunday Morning.—A very fine, clear frost. I stitched up The
Pedlar; wrote out Ruth; read it with the alterations, then wrote
Mary H. Read a little German, ... and in came William, I did not
expect him till to-morrow. How glad I was. After we had talked
about an hour, I gave him his dinner. We sate talking and happy. He
brought two new stanzas of Ruth....
Monday Morning.—A soft rain and mist. We walked to Rydale for
letters. The Vale looked very beautiful in excessive simplicity, yet, at
the same time, in uncommon obscurity. The Church stood alone—
mountains behind. The meadows looked calm and rich, bordering on
the still lake. Nothing else to be seen but lake and island....
On Friday evening the moon hung over the northern side of the
highest point of Silver How, like a gold ring snapped in two, and
shaven off at the ends. Within this ring lay the circle of the round
moon, as distinctly to be seen as ever the enlightened moon is.
William had observed the same appearance at Keswick, perhaps at
the very same moment, hanging over the Newland Fells. Sent off a
letter to Mary H., also to Coleridge, and Sara, and rewrote in the
evening the alterations of Ruth, which we sent off at the same time.
Tuesday Morning.—William was reading in Ben Jonson. He read
me a beautiful poem on Love.... We sate by the fire in the evening,
and read The Pedlar over. William worked a little, and altered it in a
few places....
Wednesday.— ... Wm. read in Ben Jonson in the morning. I read
a little German. We then walked to Rydale. No letters. They are
slashing away in Benson's wood. William has since tea been talking
about publishing the Yorkshire Wolds Poem with The Pedlar.
Thursday.—A fine morning. William worked at the poem of The
Singing Bird.54
Just as we were sitting down to dinner we heard Mr.

Clarkson's voice. I ran down, William followed. He was so finely
mounted that William was more intent upon the horse than the rider,
an offence easily forgiven, for Mr. Clarkson was as proud of it himself
as he well could be....
Friday.—A very fine morning. We went to see Mr. Clarkson off.
The sun shone while it rained, and the stones of the walls and the
pebbles on the road glittered like silver.... William finished his poem
of The Singing Bird. In the meantime I read the remainder of
Lessing. In the evening after tea William wrote Alice Fell. He went to
bed tired, with a wakeful mind and a weary body....
Saturday Morning.—It was as cold as ever it has been all winter,
very hard frost.... William finished Alice Fell, and then wrote the
poem of The Beggar Woman, taken from a woman whom I had seen
in May (now nearly two years ago) when John and he were at
Gallow Hill. I sate with him at intervals all the morning, took down
his stanzas, etc.... After tea I read to William that account of the
little boy belonging to the tall woman, and an unlucky thing it was,
for he could not escape from those very words, and so he could not
write the poem. He left it unfinished, and went tired to bed. In our
walk from Rydale he had got warmed with the subject, and had half
cast the poem.
Sunday Morning.—William ... got up at nine o'clock, but before he
rose he had finished The Beggar Boy, and while we were at
breakfast ... he wrote the poem To a Butterfly! He ate not a morsel,
but sate with his shirt neck unbuttoned, and his waistcoat open
while he did it. The thought first came upon him as we were talking
about the pleasure we both always felt at the sight of a butterfly. I
told him that I used to chase them a little, but that I was afraid of
brushing the dust off their wings, and did not catch them. He told
me how he used to kill all the white ones when he went to school
because they were Frenchmen.... I wrote it down and the other
poems, and I read them all over to him.... William began to try to
alter The Butterfly, and tired himself....

Monday Morning.—We sate reading the poems, and I read a little
German.... During W.'s absence a sailor who was travelling from
Liverpool to Whitehaven called, he was faint and pale when he
knocked at the door—a young man very well dressed. We sate by
the kitchen fire talking with him for two hours. He told us interesting
stories of his life. His name was Isaac Chapel. He had been at sea
since he was 15 years old. He was by trade a sail-maker. His last
voyage was to the coast of Guinea. He had been on board a slave
ship, the captain's name Maxwell, where one man had been killed, a
boy put to lodge with the pigs and was half eaten, set to watch in
the hot sun till he dropped down dead. He had been away in North
America and had travelled thirty days among the Indians, where he
had been well treated. He had twice swam from a King's ship in the
night and escaped. He said he would rather be in hell than be
pressed. He was now going to wait in England to appear against
Captain Maxwell. "O he's a Rascal, Sir, he ought to be put in the
papers!" The poor man had not been in bed since Friday night. He
left Liverpool at 2 o'clock on Saturday morning; he had called at a
farm house to beg victuals and had been refused. The woman said
she would give him nothing. "Won't you? Then I can't help it." He
was excessively like my brother John.
Tuesday.— ... William went up into the orchard, ... and wrote a
part of The Emigrant Mother. After dinner I read him to sleep. I read
Spenser.... We walked to look at Rydale. The moon was a good
height above the mountains. She seemed far distant in the sky.
There were two stars beside her, that twinkled in and out, and
seemed almost like butterflies in motion and lightness. They looked
to be far nearer to us than the moon.
Wednesday.—William went up into the orchard and finished the
poem. I went and sate with W. and walked backwards and forwards
in the orchard till dinner time. He read me his poem. I read to him,
and my Beloved slept. A sweet evening as it had been a sweet day,
and I walked quietly along the side of Rydale lake with quiet
thoughts—the hills and the lake were still—the owls had not begun

to hoot, and the little birds had given over singing. I looked before
me and saw a red light upon Silver How as if coming out of the vale
below,
There was a light of most strange birth,
A light that came out of the earth,
And spread along the dark hill-side.
Thus I was going on when I saw the shape of my Beloved in the
road at a little distance. We turned back to see the light but it was
fading—almost gone. The owls hooted when we sate on the wall at
the foot of White Moss; the sky broke more and more, and we saw
the moon now and then. John Gill passed us with his cart; we sate
on. When we came in sight of our own dear Grasmere, the vale
looked fair and quiet in the moonshine, the Church was there and all
the cottages. There were huge slow-travelling clouds in the sky, that
threw large masses of shade upon some of the mountains. We
walked backwards and forwards, between home and Olliff's, till I
was tired. William kindled, and began to write the poem. We carried
cloaks into the orchard, and sate a while there. I left him, and he
nearly finished the poem. I was tired to death, and went to bed
before him. He came down to me, and read the poem to me in bed.
A sailor begged here to-day, going to Glasgow. He spoke cheerfully
in a sweet tone.
Thursday.—Rydale vale was full of life and motion. The wind blew
briskly, and the lake was covered all over with bright silver waves,
that were there each the twinkling of an eye, then others rose up
and took their place as fast as they went away. The rocks glittered in
the sunshine. The crows and the ravens were busy, and the thrushes
and little birds sang. I went through the fields, and sate for an hour
afraid to pass a cow. The cow looked at me, and I looked at the
cow, and whenever I stirred the cow gave over eating.... A parcel
came in from Birmingham, with Lamb's play for us, and for C.... As
we came along Ambleside vale in the twilight, it was a grave
evening. There was something in the air that compelled me to

various thoughts—the hills were large, closed in by the sky.... Night
was come on, and the moon was overcast. But, as I climbed the
moss, the moon came out from behind a mountain mass of black
clouds. O, the unutterable darkness of the sky, and the earth below
the moon, and the glorious brightness of the moon itself! There was
a vivid sparkling streak of light at this end of Rydale water, but the
rest was very dark, and Loughrigg Fell and Silver How were white
and bright, as if they were covered with hoar frost. The moon retired
again, and appeared and disappeared several times before I reached
home. Once there was no moonlight to be seen but upon the island-
house and the promontory of the island where it stands. "That needs
must be a holy place," etc. etc. I had many very exquisite feelings,
and when I saw this lofty Building in the waters, among the dark
and lofty hills, with that bright, soft light upon it, it made me more
than half a poet. I was tired when I reached home, and could not sit
down to reading. I tried to write verses, but alas! I gave up,
expecting William, and went soon to bed.
Friday.—A very rainy morning. I went up into the lane to collect a
few green mosses to make the chimney gay against my darling's
return. Poor C., I did not wish for, or expect him, it rained so....
Coleridge came in. His eyes were a little swollen with the wind. I was
much affected by the sight of him, he seemed half-stupefied. William
came in soon after. Coleridge went to bed late, and William and I
sate up till four o'clock. A letter from Sara sent by Mary. They
disputed about Ben Jonson. My spirits were agitated very much.
Saturday.— ... When I awoke the whole vale was covered with
snow. William and Coleridge walked.... We had a little talk about
going abroad. After tea William read The Pedlar. Talked about
various things—christening the children, etc. etc. Went to bed at 12
o'clock.
Sunday.—Coleridge and William lay long in bed. We sent up to
George Mackareth's for the horse to go to Keswick, but we could not
have it. Went with C. to Borwick's where he left us. William very

unwell. We had a sweet and tender conversation. I wrote to Mary
and Sara.
Monday.—A rainy day. William very poorly. 2 letters from Sara,
and one from poor Annette. Wrote to my brother Richard. We talked
a good deal about C. and other interesting things. We resolved to
see Annette, and that Wm. should go to Mary. Wm. wrote to
Coleridge not to expect us till Thursday or Friday.
Tuesday.—A mild morning. William worked at The Cuckoo poem. I
sewed beside him.... I read German, and, at the closing-in of day,
went to sit in the orchard. William came to me, and walked
backwards and forwards. We talked about C. Wm. repeated the
poem to me. I left him there, and in 20 minutes he came in, rather
tired with attempting to write. He is now reading Ben Jonson. I am
going to read German. It is about 10 o'clock, a quiet night. The fire
flickers, and the watch ticks. I hear nothing save the breathing of my
Beloved as he now and then pushes his book forward, and turns
over a leaf....
Wednesday.—It was a beautiful spring morning, warm, and quiet
with mists. We found a letter from M. H. I made a vow that we
would not leave this country for G. Hill.55
... William altered The
Butterfly as we came from Rydale....
Thursday.— ... No letter from Coleridge.
Friday.— ... William wrote to Annette, then worked at The
Cuckoo.... After dinner I sate 2 hours in the orchard. William and I
walked together after tea, to the top of White Moss. I left Wm. and
while he was absent I wrote out poems. I grew alarmed, and went
to seek him. I met him at Mr. Olliff's. He has been trying, without
success, to alter a passage—his Silver How poem. He had written a
conclusion just before he went out. While I was getting into bed, he
wrote The Rainbow.
Saturday.—A divine morning. At breakfast William wrote part of
an ode.... We sate all day in the orchard.

Sunday.—We went to Keswick. Arrived wet to the skin....
Monday.—Wm. and C. went to Armathwaite.
Tuesday, 30th March.—We went to Calvert's.
Wednesday, 31st March.— ... We walked to Portinscale, lay upon
the turf, and looked into the Vale of Newlands; up to Borrowdale,
and down to Keswick—a soft Venetian view. Calvert and Wilkinsons
dined with us. I walked with Mrs. W. to the Quaker's meeting, met
Wm., and we walked in the field together.
Thursday, 1st April.—Mrs. C, Wm. and I went to the How. We
came home by Portinscale.
Friday, 2nd.—Wm. and I sate all the morning in the field.
Saturday, 3rd.—Wm. went on to Skiddaw with C. We dined at
Calvert's....
Sunday, 4th.—We drove by gig to Water End. I walked down to
Coleridge's. Mrs. Calvert came to Greta Bank to tea. William walked
down with Mrs. Calvert, and repeated his verses to them....
Monday, 5th.—We came to Eusemere. Coleridge walked with us to
Threlkeld....
* * * * * *
Monday, 12th.— ... The ground covered with snow. Walked to
T. Wilkinson's and sent for letters. The woman brought me one from
William and Mary. It was a sharp, windy night. Thomas Wilkinson
came with me to Barton, and questioned me like a catechiser all the
way. Every question was like the snapping of a little thread about my
heart. I was so full of thought of my half-read letter and other
things. I was glad when he left me. Then I had time to look at the
moon while I was thinking my own thoughts. The moon travelled
through the clouds, tinging them yellow as she passed along, with

two stars near her, one larger than the other. These stars grew and
diminished as they passed from, or went into, the clouds. At this
time William, as I found the next day, was riding by himself between
Middleham and Barnard Castle....
Tuesday, 13th April.—Mrs. C. waked me from sleep with a letter
from Coleridge.... I walked along the lake side. The air was become
still, the lake was of a bright slate colour, the hills darkening. The
bays shot into the low fading shores. Sheep resting. All things quiet.
When I returned William was come. The surprise shot through me....
* * * * * *
Thursday, 15th.—It was a threatening, misty morning, but mild.
We set off after dinner from Eusemere. Mrs. Clarkson went a short
way with us, but turned back. The wind was furious, and we thought
we must have returned. We first rested in the large boathouse, then
under a furze bush opposite Mr. Clarkson's. Saw the plough going in
the field. The wind seized our breath. The lake was rough. There
was a boat by itself floating in the middle of the bay below Water
Millock. We rested again in the Water Millock Lane. The hawthorns
are black and green, the birches here and there greenish, but there
is yet more of purple to be seen on the twigs. We got over into a
field to avoid some cows—people working. A few primroses by the
roadside—woodsorrel flower, the anemone, scentless violets,
strawberries, and that starry, yellow flower which Mrs. C. calls pile
wort. When we were in the woods beyond Gowbarrow Park we saw
a few daffodils close to the water-side. We fancied that the sea had
floated the seeds ashore, and that the little colony had so sprung up.
But as we went along there were more and yet more; and at last,
under the boughs of the trees, we saw that there was a long belt of
them along the shore, about the breadth of a country turnpike road.
I never saw daffodils so beautiful. They grew among the mossy
stones about and above them; some rested their heads upon these
stones, as on a pillow, for weariness; and the rest tossed and reeled

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